1. Introduction
It is well known that hidden symmetry properties, related to symplectic, differential-geometric, differential-algebraic (D-A) or analytical structures of nonlinear Hamiltonian dynamical systems on functional manifolds, such as an infinite hierarchy of conservation laws and compatible Poissonian structures, often give rise to their Lax integrability. This fact was extensively worked out by many researchers during the past half century and a very powerful so called inverse Lie-algebraic orbit method [1,3-6] of constructing hierarchies of a priori Lax integrable nonlinear dynamical systems was devised. A related direct problem of retrieving these hidden intrinsic symmetries for a priori given well posed nonlinear dynamical system, which are suspected to be Lax integrable, proved to be a very complicated task, whose solution is still far from being solved. Among different approaches to coping with it one can mention, for instance, the classical Kowalewskaya-Painlevé method and its modifications, the Mikhaylov-Shabat [7] recursion operator method, based on analyzing the Lie-Backlund symmetries and some other techniques, which appeared to be reasonably effective in diverse applications, especially for classifying nonlinear integrable dynamical systems possessing special structure. Recently, when studying integrability properties of infinite so called Riemann type hydrodynamical hierarchies, a new direct approach to testing the Lax integrability of a priori given nonlinear dynamical systems with special structure, based on treating the related symplectic and differential-algebraic structures of differentiations, was suggested [8] and devised in [9]. By means of this technique the direct integrability problem was effectively reduced to the classical one of finding the corresponding compatible representations in suitably constructed differential rings.
Concerning the inverse Lie-algebraic orbit method, as its name suggests, it consists [1,3,6,10-12] in studying invariant orbits of the coadjoint group 
action on a specially chosen element 
where 
is the conjugate space to the Lie algebra 
of a suitably chosen, in general formal, group 
In other words, the main Lie-algebraic essence of this approach consists in considering functional invariance and related symplectic properties of these extended orbits in 
generated by the given element 
and inherited from the standard Lie algebra structure of 
From this point of view, subject to this extension scheme of constructing a priori Lax integrable dynamical systems, it was natural to search for another way of constructing such systems, but based on a suitably chosen reduction construction of the corresponding coadjoint group 
action on the general element 
Happily, in modern symplectic geometry such a reduction method was well developed many years ago by Marsden and Weinstein [13,14] and effectively applied to studying integrability properties of some nonlinear dynamical systems [15,16] on finite-dimensional symplectic manifolds. Thus, a next step, consisting in developing this Marsden-Weinstein reduction method and applying it to the case of infinite-dimensional dynamical systems on functional manifolds, was quite natural and effectively realized in [17]. The latter, in particular, made it possible to substantially generalize results of [18] and apply them to studying a new physically feasible and important model in modern quantum physics. As all of the topics, mentioned above and recently studied in our publications, are closely connected to each other, we tried in this work to review those main essentially used analytical, Lie-algebraic and differential-algebraic structures which proved to be algorithmically effective for studying Lax integrability of nonlinear dynamical systems on functional manifolds.
As an important example of applying these recently devised techniques, a new generalized Riemann type hydrodynamic system is studied by means of a novel combination of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are constructed. Also analyzed is the complete Lax integrability of the important (for applications) Ostrovsky-Vakhnenko Equation, studied by means of symplectic, gradient-holonomic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax representations and related infinite hierarchies of conservation laws are also presented.
2. Lax Integrability via Marsden-Weinstein Reduction and the AKS-BK and R-Matrix Approaches
Loop Group, Canonically Symplectic Manifold and Hamiltonian Action
As it is well-known [1,6,13,14], the most popular canonically symplectic manifolds are supplied by cotangent spaces 
to some “coordinates” phase spaces
, which can often possess additional symmetry properties. If this symmetry can be identified with a Lie group
action on the phase space 
and its natural extension on the whole manifold 
proves to be symplectic and even more, Hamiltonian, the Marsden-Weinstein reduction method [10,13] makes it possible to construct new Hamiltonian flows on the smaller invariant reduced phase space 
subject to the group invariant constraint 
for some specially chosen element
, where 
is the related momentum mapping on the symplectic manifold 
and 
is the adjoint space to the Lie algebra 
of the group Lie
.
As the corresponding Hamiltonian flows on the reduced phase space 
often possess very interesting properties important for applications in many branches of mathematics and physics, they were topics of many investigations during the past decades. As a result of our interest in the mathematical properties of the Lax integrable flows, we observed that their modern Lie algebraic description by means of the Hamiltonian group action classical Lie-Poisson-Adler-Kostant-Berezin-Kirillov scheme on the adjoint space 
to the Lie algebra 
of a suitably chosen group 
is a natural consequence of applying the Marsden-Weinstein reduction method to the canonical symplectic phase space
. The basis space
, has to be a specially chosen Lie algebra 
with the naturally related Hamiltonian group 
action on the symplectic phase space 
Moreover, such classical integrability theory ingredients as 
-structures [19] and the related commutation properties of the related transfer matrices are also naturally retrieved from the Marsden-Weinstein reduction method via the scheme specified above.
Consider a complex matrix Lie group 
, its Lie algebra
, and a related [1,4,6] formal loop group 
of G-valued functions on the circle
, meromorphically depending on the complex parameter
. Its Lie algebra 
can be viewed as the completion
(1)
By the standard procedure [1,10] one can construct the centrally extended current algebra
, on which the adjoint loop group 
-action is defined: for any 
(2)
Here 
and 
is the following nondegenerate symmetric scalar product on
:
(3)
for any
. The scalar product (3) is ad-invariant, that is
(4)
for any elements 
and 
Define now the canonically symplectic phase space
with the corresponding Liouville 1-form on 
(5)
whose exterior derivative gives the symplectic structure on the functional manifold
:
(6)
Similarly to (2) one can naturally extend the group 
-action on the whole phase space
, having
(7)
for any 
and 
as the corresponding co-adjoint action of the current group 
to the adjoint linear space 
The following lemma is almost selfevident.
Lemma 1 The 
-group action (2) and (7) on the symplectic phase space 
is symplectic and Hamiltonian.
It is easy to check that the canonical Liouville 1-form (5) on the manifold 
is 
-invariant:
(8)
From (8), owing to the expression (6), one obtains the symplectic form invariance
(9)
for any element 
To define the Hamiltonian G-action on the symplectic manifold M we take the group flow 
for
, 
, and find the generated vector field 
on the phase space
:
(10)
by a Hamiltonian function 
owing to the canonical relationship 
(11)
As a consequence of (11), one obtains
(12)
for any point 
From (12) it follows that
(13)
is linear with respect to the generator element 
This means that the loop group 
action on the symplectic manifold 
is Hamiltonian by definition [12,13].
The corresponding mapping 
where
(14)
is called the momentum mapping [10,12,13] which can be constrained to be fixed for further applications to the phase space 
in the Marsden-Weinstein reduction procedure [13].
Let us describe in detail the related symplectic structure on the 
-level submanifold
(15)
for a fixed element 
As a more natural case we take that 
The corresponding isotropy group
, as 
holds for any element 
To proceed further, we need some additional properties of the submanifold 
which we describe next.
3. Marsden-Weinstein Reduction and Poisson Brackets
In this section we shall be interested in describing the submanifold 
parameterized by the points of the reduced phase space
. It is known [13,14] that this parametrization uniquely determines the points 
which are invariant with respect to the appropriate loop group 
action (2) and (7). The last property makes it possible [10,13,14,20] to define on the phase space 
the reduced nondegenerate symplectic structure on the phase space 
by means of the appropriate symplectic structure on the submanifold
. Let us consider the point 
where the elements
, 
according to the definition (15), satisfy the differential expressions:
(16)
for all 
Consider now a Hamiltonian vector field 
on the submanifold 
generated by the element 
owing to the expressions
(17)
From (17) it follows that the equality 
holds on the reduced phase space 
Let us also compute the evolution of the element 
with respect to this vector field 
on 
(18)
coinciding with the a priori assumed condition 
for any 
Define similarly a vector field 
on the reduced phase space 
generated by the Lie algebra element 
depending on the basis element 
such that
(19)
This, in particular, means that the flows 
and 
on the reduced phase space 
possess the countable set 
of conservation lows, where by definition, the element 
satisfies for a given element 
the determining equation
(20)
for all 
From the Equation (20) one easily finds that upon the reduced phase space 
(21)
Thus, from the 
-evolution (21) of the parameter 
one finds that the constraint
(22)
holds on the reduced phase space 
subject to the vector field 
generated by the element 
Moreover, as it is easy to observe, these two vector fields 
and 
on the reduced phase space 
commute:
(23)
This is very promising, since the condition (23) results in a differential relationships on the components of the reduced matrix 
for which the related evolution equation
(24)
and the differential Equation
(25)
for the matrix 
are compatible. These Equations (24) and (25) realize the well-known [1,4-6,10,12] generalized Lax spectral problem, allowing to integrate the above differential relationships by means of either the inverse scattering or the spectral transform methods [1,4,5,21] and algebraic geometry methods [4,5], or their modern generalizations [6].
To make this aim more constructive, it is necessary to describe the evolution of the vector field 
on the reduced phase space 
in more detail subject to its dependence on the phase space element 
Taking into account that the vector fields 
and 
satisfy the commutation condition (23) on the reduced manifold
, we will apply Marsden-Weinstein reduction theory to our symplectic manifold 
with the fixed value of the moment mapping 
for computing the Poisson bracket
(26)
of the functions 
and 
on the reduced phase space 
for arbitrary 
It can be shown [10,20,22] that this Poisson bracket on 
in general is
(27)
where, by definition, the mappings 
denote the solutions to the relationship
(28)
which holds for all 
The functions 
should be extended to the whole phase space 
in such a way that their restrictions on the submanifold 
are 
-invariant.
To apply the Marsden-Weinstein reduction, we will take into account that, by definition, there exists a group element 
such that for arbitrarily chosen 
the expression
(29)
holds and satisfies the normalization condition 
. By considering the function
(30)
one can observe that 
and, by construction, it is 
-invariant. This means that 
for any 
In fact, for any 
(31)
where we made use of the property 
This holds owing to the definitions (29) and (7):
(32)
giving rise to relationship 
for any 
and 
Returning to the Poisson bracket (27), we can replace the functions 
and 
with their
-invariant extensions 
Before calculating the corresponding Poisson bracket
(33)
where 
is the vector field generated on 
by the element 
we need to calculate the action 
for any element 
Just as with the calculations from [22], one finds that on the submanifold 
(34)
Thus, on the reduced phase space 
the general expression (34) implies
(35)
Therefore, the Poisson bracket (33), in view of the relationships 
and (35), becomes
(36)
where we take into account that owing to (28) and (35), the expression
Now one can rewrite the Poisson bracket (36) as
(37)
where, by definition, we have introduced the classical 
-matrix structure in the Lie algebra
:
(38)
where 
and the linear homomorphism 
is defined as
(39)
The mapping (39) should satisfy [23] the well-known condition
(40)
for any 
and 
Now it is useful to recall that the mapping 
satisfies the relationship (29), which implies [24] the following differential expression
(41)
for any 
where 
is the derivative mapping depending on the chosen reduction 
The mapping (39) satisfies an additional relationship, which can be obtained from the group 
-action on the element 
(42)
following naturally from (29). Differentiation of (42) with respect to 
at the point 
gives rise to
(43)
for an arbitrary 
Moreover, since the matrix (42) satisfies the relationship (20), its differentiation with respect to 
yields the differential expression:
(44)
which holds for any
. The above results can be formulated as the following proposition.
Proposition 1 The Poisson bracket (26) on the reduced phase space 
represented as a 
-structure (37) on the linear space
, naturally generated by the gauge transformation (29), which reduces the arbitrary element 
to the element 
is uniquely defined on 
As a consequence of representation (37) we find that there exists an another infinite hierarchy of mutually commuting functionals with respect to the Poisson bracket on the phase space
. The latter follows from the tensor form of the Poisson bracket (26) in the space 
(45)
which holds for arbitrary 
and where 
denotes the tensor form of the 
- structure 
The trace operation in (45) causes the Poisson bracket to vanish on the phase space 
for the functionals 
and 
for arbitrary 
4. Monodromy, R-Structure and Lie-Poisson Brackets
Next we analyze possible forms of the 
-mapping (39) as a function on the reduced phase space 
Since 
is constant, its value for convenience is set at 
Thus, taking into account the definition (39), the determining 
-structure Equation (41) takes the form:
(46)
for any element 
Let us consider the linear matrix Equation
(47)
where 
with Cauchy data at a point 
(48)
The corresponding normalized monodromy matrix
(49)
for 
and arbitrary 
satisfies
(50)
exactly coinciding with (20). Thus, if by the co-adjoint transformation (7) this chosen matrix 
is transformed into the matrix 
then the corresponding monodromy matrix of (24) transforms into the monodromy matrix of (47), which satisfies (50).
In view of the relationships (47), (48) and (50), one can recalculate the Poisson bracket (37) as
(51)
for arbitrary 
and 
It yields the following tensor expression for the reduced phase space 
(52)
where
, 
and, by definition,
(53)
The local functional matrices
satisfy the antisymmetry property:
(54)
for all 
and the permutation operator 
acts as 
for any 
Just as in the calculation from [1,25,26] one obtains from (53) that
(55)
where the matrix 
for all 
depends only on 
The expression (55) allows the very compact representation
(56)
if the tensor 
-matrix 
satisfies for 
and 
the differential relationship
(57)
If we define the mapping 
as
(58)
for any
, then the relationship (57) can be easily presented in the following operator form:
(59)
which holds for any
, where we denoted
(60)
The result (59) can be used for rewriting the Poisson bracket (56) as
(61)
where
and we defined the gradients 
and 
in the standard way as
(62)
for any smooth functional 
and
.
It is easy to observe that under the antisymmetry condition 
the right-hand side of (61) equals the Lie-Poisson bracket [1,2,4,6,10] for the functionals
and 
Here the adjoint space
is with respect to a new commutator structure 
on the centrally extended Lie algebra 
for any 
with commutator
(63)
In (63) the classical 
-structure on the Lie algebra
under some conditions on the mapping 
can generate on 
a new Lie structure (which it must not).
The above results can be formulated as follows.
Proposition 2 The Marsden-Weinstein reduced canonical Poisson structure on the phase space 
for the monodromy matrix 
exactly coincides with the corresponding classical Lie-Poisson AKS-bracket on the centrally extended basis Lie algebra 
subject to the 
-structure (63) when it is antisymmetric.
If the antisymmetry property for the mapping 
does not hold, the generated Lie-Poisson type bracket on the functional space 
can be, owing to (61), defined as follows: for any 
the bracket
(64)
where the generalized 
-structure 
on 
is given by the expression (60).
5. D-Structure and Generalized R-Structure
As stated above, the reduced Poisson bracket on the phase space 
is
(65)
where for any 
the corresponding 
-structure on the Lie algebra 
is defined by the classical expression (38) and the mapping (39). It is natural to assume that there exists a relationship between the D-structure 
and the R-structure 
described above in Section 3.
Assume, for brevity, that the 
-structure (58) is antisymmetric, that is 
Then it is easy to check that the following algebraic relationship
(66)
holds for any 
In fact, (56) is equivalent to
(67)
Now, substituting (66) into (37), one obtains that
(68)
which coincides exactly with (67).
It is convenient to rewrite the operator relationship (46) in the tensor form as
(69)
where the tensor
, owing to the action (66), equals
(70)
Substituting the expression (70) into the Equation (69) and taking into account the determining Equation (57)
(71)
one obtains the relationship for the tensor
:
(72)
This makes two 
- and 
-structures on the Lie algebra 
compatible. Observe that the 
-structure (66) is not antisymmetric even though the 
-structure was assumed to be antisymmetric. Concerning the 
-structure determining Equation (69) one can anticipate that a study of its solutions would describe a set of nonlinear dynamical systems on the reduced phase space 
possessing an infinite hierarchy of mutually commuting conservation laws.
6. Generalized Riemann Systems: Lax Integrability and D-A Structures
6.1. Setting the Problem
Recently, new mathematical approaches based on differential-algebraic [27-31] and differential geometric methods and techniques, were applied in [8,32,33] for studying the Lax integrability of nonlinear differential equations of the Korteweg-de Vries and Riemann type. In particular, many analytical studies [32,34-40] have been devoted to finding the corresponding Lax representations of the infinite Riemann type hydrodynamical hierarchy, suggested recently by M. Pavlov and D. Holm in the form
(73)
where the differentiation 
and 
It was found that the related dynamical system
(74)
defined on a 
-periodic infinite-dimensional smooth functional manifold 
possesses [8,37] for an arbitrary integer 
a suitable Lax representation
(75)
with 
being a complex spectral parameter and 
and matrices
Here, by definition, 
and the differentiations
(76)
satisfy on the manifold 
the following commutation relationship:
(77)
In particular, for 
the following result [8,41] was recently obtained.
Proposition 3 The Lax representation for the generalized Riemann type hydrodynamical system
(78)
is given for any arbitrary 
by a set of linear compatibility equations (see Equation (79))where 
and 
is an arbitrary complex parameter. Moreover, the relationships (79) realize a linear matrix representation of the commutator condition (77).
In our work we study the complete integrability of a new dispersive Riemann type hydrodynamic flow
(80)
on a 
-periodic functional manifold
where 
is an arbitrary natural number, the vector
the differentiations
(79)
satisfy as above the Lie-algebraic commutator relationship (77) and 
is an evolution parameter. This system can be considered as a slight generalization of the dispersive Riemann hydrodynamic system (73), extensively studied by means of different mathematical tools in [8,9,32,35,37,41]. For the case 
it is well known [10,12] that the system (80) is a smooth Lax integrable bi-Hamiltonian flow on the 
-periodic functional manifold 
whose Lax representation is given by the compatible linear system
(81)
where 
and 
is an arbitrary spectral parameter.
For 
the dynamical system (80) is equivalent to that on a 
-periodic functional manifold
for a vector 
(82)
This can be easily rewritten by means of the change of variables 
as that on a 
-periodic functional manifold
for a vector 
(83)
or in the form of the flow
(84)
defining a standard smooth dynamical system on the infinite-dimensional functional manifold 
where 
is the corresponding vector field on 
We succeeded in proving the following result based on symplectic gradient-holonomic and differential algebraic tools.
Proposition 4 The Riemann type hydrodynamic flow (97) is a bi-Hamiltonian dynamical system on the functional manifold 
with respect to two compatible Poissonian structures 
(85)
possessing an infinite hierarchy of mutually commuting conservation laws and a non-autonomous Lax representation of the form (see Equation (86)).
where 
is an arbitrary spectral parameter and 
We demonstrate the effectiveness of the devised differential-algebraic tools and methods by means of application to the very interesting [42-47] nonlinear Ostrovsky-Vakhnenko hydrodynamic equation
(87)
on the 
-periodic functional manifold
subject to which the following proposition is proved.
Proposition 5 The Ostrovsky-Vakhnenko dynamical system (79) allows the standard differential Lax representation and defines on the functional manifold 
an integrable bi-Hamiltonian flow with two compatible Poisson structures. In particular, this dynamical system possesses an infinite hierarchy of mutually commuting nonlocal conservation laws.
In particular, we construct by means of the differential-algebraic tools a differential Lax representation, coinciding with that found in [43], and given in the equivalent matrix Zakharov-Shabat form as
(88)
where matrices (see Equation (89))
(86)
(89)
and 
is an arbitrary spectral parameter. We also find a related pair of compatible polynomial Poissonian structures
(90)
with 
and 
with respect to which the Ostrovsky-Vakhnenko hydrodynamic equation (87) is equivalent to a suitable bi-Hamiltonian flow on the functional manifold 
Also analyzed by means of the devised differential-algebraic tools is the Lax integrability of the interesting generalized Ostrovsky-Vakhnenko type system of evolution equations
(91)
on a 
-periodic functional manifold
6.2. Integrability of a Generalized Riemann Hydrodynamic System
6.2.1. New Generalization of the Riemann Hydrodynamic Hierarchy
In this section we shall study the complete integrability of the dispersiveness Riemann type hydrodynamic flow (80)
(92)
on a 
-periodic functional manifold
where 
is an arbitrary natural number, the vector
the differentiations
satisfy, as above, the Lie-algebraic commutator relationship (77) and 
is an evolution parameter. The system can be considered as a slight generalization of the dispersiveness Riemann hydrodynamic system suggested recently by M. Pavlov and D. Holm in the form
(93)
for 
and extensively studied in [8,9,32,35,37,41], where it was proved that it is a Lax integrable bi-Hamiltonian flow on the manifold 
and possesses an infinite hierarchy of mutually commuting dispersive Lax integrable Hamiltonian flows.
For 
it is well known [10,12] that the system (92) is a smooth Lax integrable bi-Hamiltonian flow on the 
-periodic functional manifold 
whose Lax representation is given by the compatible linear system
(94)
where 
and 
is an arbitrary spectral parameter.
Our focus here is an investigation of the Lax integrability of the Riemann type hydrodynamic system (92) for 
on a 
-periodic functional manifold 
for a vector
. We treat this problem in the following extended form:
(95)
The flow (95) can be recast as a one on a 
-periodic functional manifold 
for a vector 
as
(96)
where, for further convenience, we have made the change of variables: 
We will also use the form of the flow (96):
(97)
defining a standard smooth dynamical system on the infinite-dimensional functional manifold 
where 
is the corresponding vector field on 
In the sequel, we shall prove by means of gradientholonomic and differential algebraic tools Proposition 4, stating the Lax integrability of the dynamical system (97), in particular, we will devise an effective approach for constructing its exact Lax representation and related compatible Poissonian structures.
6.2.2. Symplectic Gradient-Holonomic Integrability Analysis: Poissonian Structure on M3
Our first steps in proving Proposition 4 are fashioned using the symplectic gradient-holonomic method, which takes us a long way towards the desired result.
By employing the symplectic gradient-holonomic approach [10,12,48] to study the integrability of smooth nonlinear dynamical systems on functional manifolds, one can find a set of conservation laws for (97) by constructing analytical solutions 
to the functional Lax gradient Equation:
(98)
where 
is a suitable Lagrangian functional and the linear operator
is the adjoint with respect to the standard convolution 
on 
of the Fréchet derivative of a nonlinear mapping
; namely,
(99)
The Lax gradient Equation (98) can be, owing to (80), rewritten as
(100)
where the matrix operator is
(101)
The first vector elements
(102)
as can be easily checked, are solutions of the functional Equation (100). From an application of the standard Volterra homotopy formula
(103)
one finds the conservation laws for (80); namely,
(104)
It is now quite easy, making use of the conservation laws (104), to construct a Poissonian structure 
for the dynamical system (97). If we use the representations
(105)
it follows that the vector 
satisfies the Lax gradient Equation (100):
(106)
where the Lagrangian function 
Thus, based on the inverse co-symplectic functional expression
(107)
one readily obtains the linear co-symplectic operator on the manifold 
(108)
which is the corresponding Poissonian operator for the dynamical system (80). It is also important to observe that the dynamical system (80) is a Hamiltonian flow on the functional manifold 
with respect to the Poissonian structure (108).
(109)
6.2.3. Poissonian Structure on 
In what follows, we shall find it convenient to construct other Poissonian structures for dynamical system (95) on the manifold 
rewritten in the equivalent form
(110)
where 
is the corresponding vector field on 
To proceed, we need to obtain additional solutions to the Lax gradient Equation (100) on the functional manifold 
(111)
where the matrix operator is
(112)
and which we may rewrite in the component form
(113)
where the vector
As a simple consequence of (113), one obtains the following system of differential relationships:
(114)
Here we have defined
and made use of the commutator relationship for differentiations 
and 
(115)
which holds for the function 
where 
It therefore follows that after solving the first equation of system (114), and one can recursively solve the remaining two equations. In particular, it is easy to see that the three vector elements
(116)
are solutions of the system (114). The first two elements of (116) lead to the Volterra symmetric vectors
entailing the trivial conservation laws
The third element of (116) gives rise to the Volterra asymmetric vector 
entailing the following inverse co-symplectic functional expression:
(117)
Correspondingly, the Poissonian operator
is
(118)
subject to which the following Hamiltonian representation
(119)
holds on the manifold
.
6.2.4. Hamiltonian Integrability Analysis
Next, we return to our integrability analysis of the dynamical system (97) on the functional manifold 
It is easy to recalculate the form of the Poissonian operator (118) on the manifold 
to that acting on the manifold 
giving rise to the second Hamiltonian representation of (97):
(120)
where 
is the corresponding Poissonian operator. As a next important point, the Poissonian operators (108) and (118) are compatible [1,3,10,12] on the manifold
; that is, the operator pencil 
is also Poissonian for arbitrary 
As a consequence, any operator of the form
(121)
for all 
is Poissonian on the manifold
. Using now the homotopy formula (103) and recursion property of the Poissonian pair (109) and (118), it is easy to construct the related infinite hierarchy of mutually commuting conservation laws
(122)
for the dynamical system (97), where 
and
is the corresponding recursion operator, which satisfies the so called associated Lax commutator relationship
(123)
In the course of the above analysis and observations, we have proved the following result.
Proposition 6 The Riemann hydrodynamic system (97) is a bi-Hamiltonian dynamical system on the functional manifold 
with respect to the compatible Poissonian structures 
(124)
and possesses an infinite hierarchy of mutually commuting conservation laws (122).
Concerning the existence of an additional infinite and parametrically 
-ordered hierarchy of conservation laws for the dynamical system (92), it is instructive to consider the dispersive nonlinear dynamical system
(125)
By solving the corresponding Lax equation
(126)
for an element 
in a suitably chosen asymptotic form, one can construct an infinite ordered hierarchy of conservation laws for (92), which we will not delve into here. This hierarchy and the existence of an infinite and parametrically 
-ordered hierarchy of conservation laws for the Riemann type dynamical system (92) provided compelling indications that it is completely integrable in the sense of Lax on the functional manifold
. We shall study the complete integrability in the next section using rather powerful differentialalgebraic tools that were devised recently in [8,9,37].
6.3. D-A Integrability Analysis for 
Consider a polynomial differential ring
generated by a fixed functional variable 
and invariant with respect to two differentiations 
and 
that satisfy the Lie-algebraic commutator relationship (77) together with the constraint (96) expressed in the differential-algebraic functional form
(127)
Since the Lax representation for the dynamical system (97) can be interpreted [8,10] as the existence of a finite-dimensional invariant ideal 
realizing the corresponding finite-dimensional representation of the Lie-algebraic commutator relationship (127), this ideal can be constructed as
(128)
where 
and 
is an arbitrary real parameter. To find finite-dimensional representations of the 
- and 
-differentiations, it is necessary [8] first to find the 
-invariant kernel 
and next to check its invariance with respect to the 
-differentiation. It is easy to show that
(129)
where the matrix 
is given as
(130)
To obtain the corresponding representation of the 
-differentiation in the space 
it suffices to find a matrix 
that
(131)
for 
and the related ideal
(132)
is 
-invariant with respect to the differentiation (131). Straightforward calculations using this invariance condition then yield the following matrix
(133)
Remark 1 Simple analogs of the above differentialalgebraic calculations for the case 
lead readily to the corresponding Riemann type hydrodynamic system
(134)
on the functional manifold
, which possesses the following matrix Lax representation:
(135)
where 
is an arbitrary spectral parameter and 
As one can readily see, these differential-algebraic results provide a direct proof of Proposition 4 describing the integrability of system (97) for 
The matrices (133) are not of standard form since they depend explicitly on the temporal evolution parameter 
Nonetheless, the matrices (130) and (133) satisfy for all 
the well-known Zakharov-Shabat type compatibility condition
(136)
which follows from the Lax type relationships (129) and (131)
(137)
and the commutator condition (127). Moreover, taking into account that the dynamical system (97) has a compatible Poissonian pair (108) and (118) depending only on the variables 
and not depending on the temporal variable 
one can certainly assume that it also possesses a standard autonomous Lax representation, which can possibly be found by means of a suitable gauge transformation of (137). We plan to pursue this line of analysis in a forthcoming paper.
6.4. Integrability of Ostrovsky-Vakhnenko Equation
6.4.1. An Introduction and Problem Description
In 1998 V. O. Vakhnenko investigated high-frequency perturbations in a relaxing barotropic medium. He discovered that this phenomenon is described by a new non-linear evolution equation. Later it was proved that this equation is equivalent to the reduced Ostrovsky equation [44], which describes long internal waves in a rotating ocean. The nonlinear integro-differential Ostrovsky-Vakhnenko equation
(138)
on the real axis 
for a smooth function
, where 
is the inverse-differential operator to 
can be derived [45] as a special case of the Whitham type equation
(139)
Here the generalized kernel
and 
is an evolution parameter. Various analytical properties of (138) and related Equations were analyzed in articles [44-46,49], the corresponding Lax integrability was proved in [43].
Recently, J. C. Brunelli and S. Sakovich [42] demonstrated that the Ostrovsky-Vakhnenko Equation is a reduction of the well known Camassa-Holm Equation making it possible to construct the corresponding compatible Poisson structures for (138), but in a complicated nonpolynomial form.
In the present work we will reanalyze the integrability of Equation (138) from the gradient-holonomic [10,12,48], symplectic and formal differential-algebraic points of view. As a result, we will re-derive the Lax representation for the Ostrovsky-Vakhnenko Equation (138) and construct the related simple compatible polynomial Poisson structures and an infinite hierarchy of conservation laws.
6.4.2. Gradient-Holonomic Integrability Analysis
Consider the nonlinear Ostrovsky-Vakhnenko Equation (138) as a a nonlinear dynamical system
(140)
on the smooth 
-periodic functional manifold
(141)
where 
is the corresponding well-defined smooth vector field on 
We shall first show that the dynamical system (140) on manifold 
possesses an infinite hierarchy of conservation laws as a necessary condition for its integrability. For this we need to construct a solution to the Lax gradient equation
(142)
in the special asymptotic form
(143)
where, by definition, a linear operator
is adjoint with respect to the standard convolution 
on 
the Fréchet-derivative of a nonlinear mapping 
(144)
and, respectively,
(145)
as 
with some local functionals
on 
for all 
By substituting (143) into (142), one easily obtains the following recurrent sequence of functional relationships
(146)
for all 
modulo the Equation (140). By means of standard calculations one finds that this recurrent sequence is solvable and
(147)
and so on. It is easy check that all of functionals
(148)
are conservation laws on the manifold
, that is 
for 
with respect to the dynamical system (140). For instance, if 
one obtains:
(149)
and
(150)
since, owing to the constraint (141), the integrals
The result above suggests that the dynamical system (140) on the functional manifold 
is an integrable Hamiltonian system.
First, we will show that this dynamical system is a Hamiltonian flow
(151)
with respect to some Poisson structure
and a Hamiltonian function 
Using on the standard symplectic techniques [1,3,10,12], consider the conservation law (149) and present it in the scalar “momentum” form
(152)
with the co-vector 
and calculate the corresponding co-Poissonian structure
(153)
or the Poissonian structure
(154)
This operator 
is really Poissonian for (140) since the following determining symplectic condition
(155)
holds for the Lagrangian function
(156)
As a result of (155), one readily infers that
(157)
where the Hamiltonian function
(158)
is an additional conservation law of the dynamical system (140). Thus, one can formulate the following proposition.
Proposition 7 The Ostrovsky-Vakhnenko dynamical system (140) possesses an infinite hierarchy of nonlocal, in general, conservation laws (148) and is a Hamiltonian flow (157) on the manifold 
with respect to the Poissonian structure (154).
Remark 2 It is useful to remark here that the existence of an infinite ordered (by 
-powers) hierarchy of conservations laws (148) is a typical property [1,3,10,12] of Lax integrable Hamiltonian systems that are simultaneously bi-Hamiltonian flows with respect to a corresponding pair of compatible Poissonian structures (cf. [50]).
As is well known [1,3,10,12], the second Poissonian structure 
on the manifold 
for (140), if it exists, can be calculated as
(159)
where a covector 
is a second solution to the determining Equation (155):
(160)
for some Lagrangian functional 
It can be easily shown by means of simple but cumbersome analytical calculations based, for example, on the asymptotic small parameter method [10,12,48] and on which we will not dwell upon here.
Instead of this, we shall apply the direct differential-algebraic approach to dynamical system (140) and reveal its Lax representation both in the differential scalar and canonical matrix Zakharov-Shabat forms. Moreover, we will construct the naturally related compatible polynomial Poissonian structures for the OstrovskyVakhnenko dynamical system (140) and generate an infinite hierarchy of mutually commuting nonlocal conservation laws.
6.5. Lax Representation and Poisson Structures: A D-A Approach
We will start by constructing of the polynomial differential ring 
generated by a fixed functional variable 
and invariant with respect to two differentiations
and 
satisfying the Lie-algebraic commutator relationship (77). Since the Lax representation for the dynamical system (140) can be interpreted [8,10] as the existence of a finitedimensional invariant differential ideal 
realizing the corresponding finite-dimensional representation of the Lie-algebraic commutator relationship (77), this ideal can be presented as
(161)
where an element 
and 
are fixed. The 
-invariance of ideal (161) will be a priori evident, if the function 
satisfies the linear differential relationship
(162)
for some coefficients 
but its 
-invariance strongly depends on the element 
which can be found from the functional relationship (142) on the element
rewritten in the following form:
(163)
From the right-hand side it follows that there exists an element 
such that
(164)
Upon substituting (164) into the left hand side of (163) one finds that
(165)
where 
for a suitably chosen density element 
As an evident result of (165) one concludes that there exists an element 
such that
(166)
Turning back to the relationships (164) and (166), one concludes that the differential representation
(167)
holds.
As a further step, we can try to realize the differential ideal (161) by means of the generating element 
defined by the relationship (167). But, as it is easy to check, this differential ideal is not finite-dimensional. So, for future calculating convenience, we will represent the element 
in the following natural factorized form:
(168)
where elements 
satisfy the adjoint pairs of the following differential relationships:
(169)
and
(170)
for some elements 
and check the finite-dimensional 
- and 
-invariance of the corresponding ideal (161) generated by the element 
Now it is easy to check by means of straightforward calculations, based on the relationship (163) and (167), that the following differential equalities
(171)
and their consequences
(172)
hold. Taking into account the independence of the sets of functional elements
and
the relationships (172) together with (168), (169) and (170) make it possible to state the following lemma.
Lemma 2 The set (161) represents a 
- and 
-invariant differential ideal in the ring 
for all 
Proof 1 This result easily follows from the fact that for 
all of the relationships (172) are compatible upon taking into account the differential expressions (168) and (170). However, for 
they are not compatible.
As a corollary of Lemma 2, in light of (161) and (170), for 
one readily finds by means of elementary calculations that the related differential ideal 
is to be invariant if the following differential Lax relationships hold:
(173)
and
(174)
where 
is an arbitrary complex parameter. Moreover, they exactly coincide with those found before in [43]. The above differential relationships (173) and (174) can be equivalently rewritten in the following matrix Zakharov-Shabat type form:
(175)
where the matrices
(176)
and 
Furthermore, it follows from the differential relationships (173) and (174) that the compatibility condition (163) gives rise to the following important relationship
(177)
where the polynomial integro-differential operator
(178)
is skew-symmetric on the functional manifold 
and comprises the second compatible Poisson structure for the Ostrovsky-Vakhnenko dynamical system (140).
Now by virtue of the recurrent relationships following from substitution of the asymptotic expansion
(179)
into (177), one can determine a new infinite hierarchy of conservations laws for dynamical system (140):
(180)
for 
where
(181)
and the recursion operator 
satisfies the standard Lax representation:
(182)
The above results can be formulated as follows.
Proposition 8 The Ostrovsky-Vakhnenko dynamical system (140) allows the standard differential Lax representation (173), (174) and defines on the functional manifold 
an integrable bi-Hamiltonian flow with compatible Poisson structures (154) and (178). In particular, this dynamical system possesses an infinite hierarchy of nonlocal conservation laws (180) defined by the gradient elements (181).
Remark 3 It should be noted that the existence of an infinite 
-powers ordered hierarchy of conservations laws (148) is a typical property [1,3,10,12] of the Lax integrable Hamiltonian systems, which are simultaneously bi-Hamiltonian flows with respect to corresponding compatible Poissonian structures.
Remark 4 It is interesting to observe that our second polynomial Poisson structure (178) differs from that obtained recently in [42], which contains rational power factors.
Making use of the differential Expressions (173) and (174), it is easy to construct a slightly different from (175) matrix Lax representation of the Zakharov-Shabat form for the dynamical system (138).
In fact, if one defines the “spectral” parameter 
and new basis elements of the invariant differential ideal (161):
(183)
then relationships (173) and (174) can be rewritten as follows:
(184)
where the matrices
(185)
coincide with those of [42,43] and satisfy the Zakharov-Shabat type compatibility condition:
(186)
Remark 5 As already mentioned above, the Lax representation (185) of the Ostrovsky-Vakhnenko dynamical system (138) was obtained in [43] by means of a suitable limiting reduction of the Degasperis-Processi equation
(187)
For convenience, let us rewrite this in the following form:
(188)
where differentiations
and 
satisfy the Lie-algebraic relationship (77). It is impressive that Equation (187) is itself a special reduction of a new Lax integrable Riemann type hydrodynamic system, proposed and studied (for
) recently in [51]:
(189)
where 
are arbitrary natural numbers. Actually, defining 
and 
from (189) one easily obtains the following dynamical system:
(190)
coinciding with the Degasperis-Processi Equation (188) if one makes the identification 
As a resultwe have proved that a function 
that satisfies, for an arbitrary
, the generalized Riemann type hydrodynamical equation
(191)
simultaneously solves the Degasperis-Processi equation (187). In particular, for 
we find that solutions to the Burgers type equation
(192)
are also solutions to the Degasperis-Processi Equation (187). This means, in particular, that the reduction procedure in [43] can also be applied to the Lax integrable Riemann type hydrodynamic system (189), giving rise to a related Lax representation for the Ostrovsky-Vakhnenko dynamical system (138).
7. Conclusions
We have considered the standard canonically symplectic phase space
, generated by the centrally extended basis manifold to be an affine loop Lie algebra 
on the circle 
Subject to the standard Hamiltonian Lie algebra 
-action on 
with respect which the symplectic structure on 
is invariant, we have constructed the corresponding momentum mapping and carried out the standard Marsden-Weinstein reduction of the manifold 
upon the reduced phase space 
endowed with the reduced Poisson bracket
. This allows to construct on the phase space 
mutually commuting vector fields which are equivalent to nonlinear dynamical systems possessing an infinite hierarchy of commuting conservation laws. Moreover, these commuting vector fields on 
realize exactly their corresponding Lax representations.
In addition, we have detailed analysis of commutation properties for the related flows on the basis manifold making it possible to define a suitable 
-structure on the Lie algebra 
intimately related to the corresponding classical 
-structure on
, generated by the reduced Poisson bracket on the phase space
. As a by-product of our analysis we proved that these 
- and 
-structures are completely equivalent to a suitably generalized classical Lie-Poisson-Adler-Kostant-SymesKirillov-Berezin structure on the adjoint space
. We also derived the determining Equation for the 
-structure, classifying the generalized Lax integrable nonlinear dynamical systems on the reduced phase space
, whose respectively defined R-structures are not necessary both antisymmetric and local, as shown in [22,24] by means of another approach. It is also worth mentioning that the reduction scheme devised in this work can be applied to the centrally extended algebra of pseudo-differential operators and affine loop algebras on the circle 
A new differential-algebraic approach, elaborated in [9] for revisiting the integrability analysis of generalized Riemann type hydrodynamical Equation (73), made it possible to prove the Lax integrability of new nonlinear Hamiltonian dynamical systems representing Riemann type hydrodynamic Equations (80), (87) and (91). In particular, the integrability prerequisites of these dynamical system, such as compatible Poissonian structures, an infinite hierarchy of conservation laws and related Lax representation have been constructed by means of both the symplectic gradient-holonomic approach [10,12,48] and innovative differential-algebraic tools devised recently [8,9,35] for analyzing the integrability of a special infinite hierarchy of Riemann type hydrodynamic systems. It is also quite clear from recent research in this area and our work in this paper that the dynamical system (80) is a Lax integrable bi-Hamiltonian flow for arbitrary integers
. This is perhaps most readily verified by means of the differential-algebraic approach, which was devised and successfully applied here for the cases 
and 3.
Making use of the differential-algebraic approach, we have also re-derived the Lax representation for the Ostrovsky-Vakhnenko Equation (138) and constructed the related simple compatible polynomial Poisson structures.
As we have seen in the course of this investigation, perhaps the most important lesson that one can derive from this approach is the following: If an investigation of a given nonlinear Hamiltonian dynamical system via the gradient-holonomic method indicates (but does not necessarily prove) that the system is Lax integrable, then its Lax representation can often be shown to exist and then successfully derived by means of a suitably constructed invariant differential ideal 
of the ring 
in accordance with the differential-algebraic approach developed here for the integrability analysis of the Riemann hydrodynamical system. Consequently, when it comes applying this lesson to the investigation of other nonlinear dynamical systems, it is natural to start with systems that are known to be Lax integrable and to try to identify and characterize those algebraic structures responsible for the existence of a related finite-dimensional matrix representation for the basic 
- and 
-differentiations in a vector space 
for some finite 
It seems plausible that if one could do this for several classes of Lax integrable dynamical systems, certain patterns in the algebraic structures may be detected that can be used to assemble a more extensive array of symplectic and differential-algebraic tools capable of resolving the question of complete integrability for many other types of nonlinear Hamiltonian dynamical systems. Moreover, if the integrability is established in this manner, the approach should also serve as a means of constructing associated artifacts of the integrability such as Lax representations and hierarchies of mutually commuting invariants. As a particular differential-algebraic problem of interest concerning these matrix representations, one can seek to develop a scheme for the effective construction of functional generators of the corresponding invariant finite-dimensional ideals 
under given differential-algebraic constraints imposed on the 
- and 
-differentiations.
We have also demonstrated here that an approach combining the gradient-holonomic method with some recently devised differential-algebraic techniques can be a very effective and efficient way of investigating integrability for a particular class of infinite-dimensional Hamiltonian dynamical systems (generalized Riemann hydrodynamical systems). But a closer look at the specific details of the approach employed here reveals, we believe, that this combination of methods can be adapted to perform effective integrability analysis of a much wider range of dynamical systems.
8. Acknowledgements
D.B. acknowledges the National Science Foundation (Grant CMMI-1029809) and A.P. and Y.P. acknowledge the Scientific and Technological Research Council of Turkey (TUBITAK/NASU-111T558 Project) for partial support of their research.