On Dirichlet, Poncelet and Abel problems

We offer connections between %Theory of elliptic curve of genius 1 applies to some problems of PDE, geometry, algebra, analysis and physics. The uniqueness of the solution of the Dirichlet problem and some another boundary value problems for the string equation inside of an arbitrary biquadratic algebraic curve is considered. It is shown that the solution is non-unique if and only if corresponding the Poncelet problem for two conics %associated with the curve has a periodic trajectory. Similarly some other problems %from different %fields of mathematics are proved to be equivalent to it. Among them there are the solvability problem of the algebraic Pell-Abel equation and an indeterminacy problem of a moment problem that generalizes well-known trigonometrical moment problem. Solvability criterions of above problems can be presented in form $\theta\in\Bbb Q$ where the number $\theta=m/n$ is connected with the concrete problem data. We demonstrate also an intimate relation of the above mentioned problems with such problems of the modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Pad\'e interpolation.