We study deformations of elliptic and hyperelliptic Riemann surfaces and of a Abelian differential of the second or third kind on the surface which preserve the periods of the differential with respect to a chosen canonical homology basis of the surface. We derive differential equations with rational coefficients governing the deformations. We apply these results to the algebra-geometric solutions to the Boussinesq equation, KdV equation, sine-Gordon equation, KP equation, the Neumann system, the Toda lattice, and $SU(N)$ Witten-Seiberg theory. The talk is based on new joint results with Vasilisa Shramchenko and:
[1] V. Dragović, V. Shramchenko, Isoharmonic deformations and constrained Schlesinger systems, arXiv: 2112.04110.
[2] V. Dragović, V. Shramchenko, Deformation of the Zolotarev polynomials and Painleve VI equations, Letters Mathematical Physics, 111, 75, 28 p. 2021.
[3] V. Dragović, V. Shramchenko, Algebro-geometric approach to an Okamoto transformation, the Painleve VI and Schlesinger equations, Annales Henri Poincare, Vol. 20, No. 4, 1121–1148, 2019.