On uniformizable representation for Abelian integrals

We show how the unformizable representation for Abelian integrals of nontrivial genera arise. The technique makes use of the famous Chudnovsky's Fuchsian linear differential equations and their relation to the sixth Painleve transcendent.Comment: Final version, 7 pages; Painleve equations and related topics (Saint-Petersburg, 2011

The sixth Painleve transcendent and uniformizable orbifolds

We consider connection between the Painleve-6 equation and explicitly uniformizable orbifoldsComment: Final version, 4 pages; Painleve equations and related topics, Saint-Petersburg, 201

Elliptic Solitons and Groebner Bases

We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gr\"obner bases method and others. New integrable potentials and corresponding solutions of the Sawada-Kotera, Kaup-Kupershmidt, Boussinesq equations and others are found.Comment: 18 pages, no figures, LaTeX'2