We prove the conjectures on dimensions and characters of some quadratic algebras stated by B.L.Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad.
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.