Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations

We consider an algorithm for constructing auto-Bäcklund transformations for finite-dimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Bäcklund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Bäcklund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.