We explicitly construct the series expansion for a certain class of solutions
to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric
solutions. We express the Taylor coefficients through some universal
combinatorial constants and find recurrence relations for them. These results
are used to obtain new formulas for the genus 0 double Hurwitz numbers. They
can also serve as a starting point for a constructive approach to the Riemann
mapping problem and the inverse potential problem in 2D.Comment: 26 page