Abel's theorem and Bäcklund transformations for the Hamilton-Jacobi equations (Q521556)

A method for constructing Bäcklund transformations for finite-dimensional Hamiltonian systems is considered. As the integration of the system is reduced to the inversion of the Abel map, with the use of equations of motion, the corresponding Abel differential equations are obtained. The Bäcklund transformation is realized by means of the equivalence relation between the roots of the Abel polynomial. In this way the Bäcklund transformations for the Lagrange top, Kowalevski top and Goryachev-Chaplygin top are constructed.