Role of discriminantly separable polynomials in integrable dynamical systems

Abstract

Discriminantly separable polynomials of degree two in each of the three variables are considered. Those polynomials are by definition polynomials which discriminants are factorized as the products of the polynomials in one variable. Motivating example for introducing such polynomials is the famous Kowalevski top. Motivated by the role of such polynomials in the Kowalevski top, we generalize Kowalevski's integration procedure on a whole class of systems basically obtained by replacing so called the Kowalevski's fundamental equation by some other instance of the discriminantly separable polynomial. We present also the role of the discriminantly separable polynomils in twowell-known examples: the case of Kirchhoff elasticae and the Sokolov's case of a rigid body in an ideal fluid.