In this article, we investigate stationary coupled Korteweg–de Vries (cKdV) systems and prove that every N$N$-field stationary cKdV system can be written, after a careful reparameterization of jet variables, as a classical separable Stäckel system in N+1$N+1$ different ways. For each of these N+1$N+1$ parameterizations, we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel 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)$(N+1)$-Hamiltonian formulation for the stationary cKdV system.

This is the second article in a suite of articles investigating relations between Stackel-type systems and Painleve-type systems. In this paper, we construct isomonodromic Lax representations for Painleve-type systems found in the previous paper by Frobenius integrable deformations of Stackel-type systems. We first construct isomonodromic Lax representations for Painleve-type systems in the so-called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painleve-type systems in the nonmagnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painleve-type. We also present isomonodromic Lax representations for all one-, two-, and three-dimensional Painleve-type systems originating in our scheme. Based on these results we propose complete hierarchies of that follow from our construction.

This is the third article in our series of articles exploring connections between dynamical systems of Stackel-type and of Painleve-type. In this article, we present a method of deforming of minimally quantized quasi-Stackel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schrodinger equations possess common, multitime solutions. As in the classical case, we obtain here both magnetic and nonmagnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps between both classes of quantum systems.

In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète 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 Stäckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Viète and Newton coordinates.

Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous 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 into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems from [14].

In this article we explicitly construct Stackel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation between separable and flat coordinates for flat Stackel systems. We also exploit the geometric structure of these systems in the obtained flat coordinates. These coordinates generalize the wellknown generalized elliptic and generalized parabolic coordinates introduced by Jacobi. (C) 2015 Elsevier Inc. All rights reserved.

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 coupled KdV 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 Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical Stackel 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 Stackel systems.

We show that with every separable classical Stäckel 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 systems 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 the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems. © 2007 Elsevier Ltd. All rights reserved.