Kowalevski top and genus-2 curves

Kowalevski's curve of genus 2 is related to two other curves arising from the solution of the Kowalevski top by the method of spectral curves in the case when the angular momentum of the top is orthogonal to the gravity vector. One is the Bobenko-Reyman-Semenov-Tian-Shansky curve of genus 2, the other is the spectral curve of the Kuznetsov-Tsiganov Lax matrix, of genus 3. The relations between the curves are given by correspondences, that is, multivalued maps, inducing isogenies of the corresponding Jacobian or Prym varieties.