Nonautonomous Integrable Systems Associated with Hurwitz Spaces in Genuses Zero and One

Briefly outlining our recent work, we construct a family of nonautonomous integrable systems (deformations of the principal chiral model) in connection with the Hurwitz spaces of meromorphic functions on the Riemann sphere, cylinder, and torus. We give differential equations describing the dependence of the critical points of the rational, elliptic, and trigonometric functions on the critical values. We outline a relation to the deformation framework of Burtzev–Mikhailov–Zakharov.