Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order $n$. He proved that if the zeros of a polyanalytic function have an accumulation point of order $n$, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution of a linear homogeneous second order differential equation of elliptic type with constant coefficients.