Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain

Abstract

Variables of separation for classical $GL(3)$ magnetic chain obtained by Sklyanin form a generic positive divisor $D$ of degree $n$ on a genus $g$ non-hyperelliptic algebraic curve. Because $n>g$ this divisor $D$ has unique representative $\rho(D)$ in the Jacobian which can be constructed by using dim$|D|=n-g$ steps of Abel's algorithm. We study properties of the corresponding chain of divisors and prove that the classical $GL(3)$ magnetic chain is a superintegrable system with dim$|D|=2$ superintegrable Hamiltonians.