A rigorous, and possibly complete analysis of the phase space picture of the tippe top solutions for all initial conditions when the top does not jump and all relations between parameters $\alpha$ and $\gamma$, is for the first time presented here. It is based on the use the Jellett's integral of motion $\lambda$ and the analysis of the energy function. Theorems about stability and attractivity of the asymptotic manifold are proved in detail. Lyapunov stability of (periodic) asymptotic solutions with respect to arbitrary perturbations is shown.

We consider a hierarchy of classical Liouville completely integrable models sharing the same (linear) $r$-matrix structure obtained through an $N$-th jet-extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of the present paper is the study of the integrable model corresponding to $N=3$, since the case $N=2$ has been considered by the authors in separate papers, both in the one-body case (Lagrange top) and in the $n$-body one (Lagrange chain). We now obtain a rigid body associated with a Lie–Poisson algebra which is an extension of the Lie–Poisson structure for the two-field top, thus breaking its semidirect product structure. In the second part of the paper we construct an integrable discretization of a suitable continuous Hamiltonian flow for the system. The map is constructed following the theory of Bäcklund transformations for finite-dimensional integrable systems developed by V.B. Kuznetsov and E.K. Sklyanin.

The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in ${\bf R}^3$. A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels.

In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton–Jacobi (H–J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov–Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton–Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel'fand–Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H–J equations. We will then show how this setting works by framing results by Veselov and Penskoï about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables.

We discuss a computer implementation of the known algorithm of finding separation coordinates for a special class of orthogonal separable systems called $L$-systems or Benenti systems.

The properties of a nonlinear deformation of the isotonic oscillator are studied. This deformation affects to both the kinetic term and the potential and depends on a parameter $\lambda$ in such a way that for $\lambda=0$ all the characteristics of of the classical system are recovered. In the second part, that is devoted to the two-dimensional case, a $\lambda$-dependent deformation of the Smorodinski–Winternitz system is studied. It is proved that the deformation introduced by the parameter $\lambda$ modifies the Hamilton–Jacobi equation but preserves the existence of a multiple separability.

We consider a point moving in an ellipsoid $a_1x_1^2+a_2x_2^2+a_3x_3^2=1$ under the influence of a force with quadratic potential $V=\frac{1}{2}(b_1x_1^2+b_2x_2^2+b_3x_3^2)$. We prove that the equations of motion of the point are meromorphically integrable if and only if the condition $b_1(a_2-a_3)+b_2(a_3-a_1)+b_3(a_1-a_2)=0$ is fulfilled.

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in $\mathbb{R}^3$. Such geodesics are either connected components of real parts of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition law for elliptic functions, and the Moser–Trubowitz isomorphism between geodesics on a quadric and finite-gap solutions of the KdV equation. For the case of 3-fold and 4-fold coverings, some explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.

Lamé and Heun functions arise via separation of the Laplace equation in general Jacobi ellipsoidal or conical coordinates. In contrast to hypergeometric functions that also arise via variable separation in the Laplace equation, Lamé and Heun functions have received relatively little attention, since they are rather intractable. Nonetheless functions of Heun type do have remarkable properties, as was pointed out in the classical book "Modern Analysis" by Whittaker and Watson who devoted an entire chapter to the subject. Unfortunately the beautiful identities appearing in this chapter have received little notice, probably because the methods of proof seemed obscure. In this paper we apply the modern operator characterization of variable separation and exploit the conformal symmetry of the Laplace equation to obtain product identities for Heun type functions. We interpret the Niven transform as an intertwining operator under the action of the conformal group. We give simple operator derivations of some of the basic formulas presented by Whittaker and Watson and then show how to generalize their results to more complicated situations and to higher dimensions.

We establish a correspondence between the SU(2) Witten–Reshetikhin–Turaev invariant for the Seifert manifold $M(p_1, p_2, p_3)$ and Ramanujan's mock theta functions.

Some abelian varieties admit several (principal) polarizations. The problem of which Jacobi varieties do, especially among the splittable, challenges transliteration from analysis to algebra. Questions and examples are provided, as well as applications to solution of certain Partial Differential Equations.