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A formalism of systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is showed that this IVP has a formal solution and within the framewrok of two particular subalgebras of the hereditary Lie algebra the explicit forms of this formal solution are presented. Finally, this formalism is applied to Korteveg-de Vries, dispersive water waves and Ablowitz-Kaup-Newell-Segur soliton hierarchies., Comment: 25 pages

In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider St\"ackel transform between two pairs of St\"ackel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between St\"ackel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems., Comment: Version accepted for publication in Special Issue 2 of Open Communications in Nonlinear Mathematical Physics. One reference added, some language issues fixed, additional explanations in Introduction

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyperelliptic type. These Miura maps induce Miura maps between St\"{a}ckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for St\"{a}ckel systems.

In this article we investigate stationary cKdV systems and prove that every $N$-field stationary cKdV system can be written, after a careful reparametrization of jet variables, as a classical separable St\"ackel system on $N+1$ different ways. For each of these $N+1$ parametrizations we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of St\"ackel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite-dimensional Miura map, which leads to an $(N+1)$-Hamiltonian formulation for the stationary cKdV system., Comment: some rearrangement, 42 pages

This is the third article in our series of articles exploring connections between dynamical systems of St\"ackel-type and of Painlev\'e-type. In this article we present a method of deforming of minimally quantized quasi-St\"ackel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schr\"odinger equations posses common, multi-time solutions. As in the classical case, we obtain here both magnetic and non-magnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps between both classes of quantum systems.

In this article we study St\"{a}ckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable St\"{a}ckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation., Comment: 18 pagages

This is the second article in a suite of articles investigating relations between St\"{a}ckel-type systems and Painlev\'{e}-type systems. In this article we construct isomonodromic Lax representations for Painlev\'{e}-type systems found in the previous paper by Frobenius integrable deformations of St\"{a}ckel-type systems. We first construct isomonodromic Lax representations for Painlev\'{e}-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlev\'{e}-type systems in the non-magnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlev\'{e}-type. We also present isomonodromic Lax representations for all one-, two- and three-dimensional Painlev\'{e}-type systems originating in our scheme. Based on these results we propose complete hierarchies of $P_{I}-P_{IV}$ that follow from our construction., Comment: 40 pages

In this paper we discuss two canonical transformations that turn St\"{a}ckel separable Hamiltonians of Benenti type into polynomial form: transformation to Vi\`ete coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of St\"{a}ckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Vi\`ete and Newton coordinates.

This article is the first one in a suite of three articles exploring connections between dynamical systems of St\"{a}ckel-type and of Painlev\'{e}- type. In this article we present a deformation of autonomous St\"{a}ckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-St\"{a}ckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-St\"{a}ckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3\,$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlev\'{e} equations $P_{I}-P_{IV}$ can be derived from our scheme., Comment: 32 pages

We construct Lax pairs for a wide class of St\"ackel systems by applying the multi-parameter St\"ackel transform to Lax pairs of a suitably chosen systems from the seed class. For a given St\"ackel system, the obtained set of non-equivalent Lax pairs is parametrized by an arbitrary function.

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-St\"ackel systems., Comment: 14 pages, no figures. We repaired Example 5 and also made some minor amendments in the text

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-St\"ackel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable St\"ackel systems. We describe the relations between Poisson algebras generated by quasi-St\"ackel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply St\"ackel transform to obtain new non-homogeneous equations of considered type.

In this paper we discuss maximal superintegrability of both classical and quantum St\"ackel systems. We prove a sufficient condition for a flat or constant curvature St\"ackel system to be maximally superintegrable. Further, we prove a sufficient condition for a St\"ackel transform to preserve maximal superintegrability and we apply this condition to our class of St\"ackel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta St\"ackel system (defined on a 2n-dimensional Poisson manifold) for which the St\"ackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable St\"ackel system., Comment: We added the journal reference and also intorduced several minor amendmends

In this article we explicitely construct transformation bewteen separable and flat coordinates for flat St\"ackel systems and exploit the structre of these systems in flat coordinates. In the elliptic case these coordinates become well known generalized elliptical coordinates of Jacobi.

In this paper we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible cKdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.

In this paper we ivestigate St\"ackel transforms between different classes of parameter-dependent St\"ackel separable systems of the same dimension. We show that the set of all St\"ackel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single St\"ackel transform. We also give an explicit formula relating solutions of two St\"ackel-related systems. These results show in particular that any two geodesic St\"ackel systems are St\"ackel equivalent in the sense that it is possible to transform one into another by a single St\"ackel transform. We also simplify proofs of some known statements about multiparameter St\"ackel transform., Comment: We clarified arguments in Section 3, Studies in Applied Mathematics 2012

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical St\"ackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding St\"ackel systems., Comment: 16 pages

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new., Comment: 15 pages

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems., Comment: 20 pages, LaTeX, no figures

We propose a general scheme of constructing of soliton hierarchies from finite dimensional St\"{a}ckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in momenta integrals of motion., Comment: 24 pages

Given a foliation S of a manifold M, a distribution Z in M transveral to S and a Poisson bivector \Pi on M we present a geometric method of reducing this operator on the foliation S along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulas that allow for an explicit calculation of the reduced Poisson bracket. Moreover, we analyse the reduction of Hamiltonian systems corresponding to the bivector \Pi., Comment: To appear in Rep. Math. Phys. LaTeX file generated by SWP 4.0

A new notion of a dual Poisson-presymplectic pair is introduced and its properties are examined. The procedure of Dirac reduction of Poisson operators onto submanifolds proposed by Dirac is in this paper embedded in a geometric procedure of reduction of dual Poisson-presymplectic pairs. The method presented generalizes those used by Marsden and Ratiu for reductions of Poisson manifolds. Two examples are given., Comment: to appear in Journal of Physics A: Mathematical and General

The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu reduction procedure. The Dirac classification of constraints on 'first-class' constraints and 'second-class' constraints is reexamined., Comment: This is a revised version of an article published in J. Nonlinear Math. Phys. vol. 10, No. 4, (2003), 451-463

Dirac deformation of Poisson operators of arbitrary rank is considered. The question when Dirac reduction does not destroy linear Poisson pencils is studied. A class of separability preserving Dirac reductions in the corresponding quasi-bi-Hamiltonian systems of Benenti type is discussed. Two examples of such reductions are given.

We perform variable separation in the quasi-potential systems of equations of the form $\ddot{q}=-A^{-1}\nabla k=-\tilde{A}^{-1}\nabla\tilde{k}${}, where $A$ and $\tilde{A}$ are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux-Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions., Comment: LaTex, 30 pages, to appear in J. Phys. A: Math. Gen

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type., Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J. Math. Phys