Dirichlet and Neumann Problems for String Equation, Poncelet Problem and   Pell-Abel Equation
  We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution. 
Vladimir P. Burskii and Alexei S. Zhedanov