From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems $H=\tfrac12 p^2+V(q)$ is explained in the first part of this review. It has a form of an effective criterion that for any given potential $V(q)$ tells whether there exist suitable separation coordinates $x(q)$ and how to find these coordinates, so that the Hamilton–Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials $V(q)$ all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics.
Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $\ddot{q}=M(q)$ admitting $n$ quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $\ddot{q}=-\nabla V(q)$. The cofactor pair Newton equations admit a Hamilton–Poisson structure in an extended $2n+1$ dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.