  We propose interconnections between some problems of PDE, geometry, algebra,
calculus and physics. Uniqueness of a solution of the Dirichlet problem and of
some other boundary value problems for the string equation inside an arbitrary
biquadratic algebraic curve is considered. It is shown that a solution is
non-unique if and only if a corresponding Poncelet problem for two conics has a
periodic trajectory. A set of problems is proven to be equivalent to the above
problem. Among them are the solvability problem of the algebraic Pell-Abel
equation and the indeterminacy problem of a new moment problem that generalizes
the well-known trigonometrical moment problem. Solvability criteria of the
above-mentioned problems can be represented in form $\theta\in\Bbb Q$ where
number $\theta=m/n$ is built by means of data for a problem to solve. We also
demonstrate close relations of the above-mentioned problems to such problems of
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.
