One-parametric families of compact Riemann surfaces over the Riemann sphere

We suggest a new method to find a polynomial uniformizing a given compact Riemann surface of genus 0. The method is in including the surface into a smooth one-parametric family of surfaces, similar to this one, and in deducing a system of ODE for critical points of the polynomials uniformizing the surfaces of the family. We also consider the case of genus one (complex tori).