On one inner theorem of uniqueness for linear elliptic second order equation with constant coefficients

We consider a linear elliptic second order differential equation. We prove that its solution $f$ is identical to zero if zeros of $f$ are condensed to two points along the non-collinear rays. We construct an example that shows that the requirement of non-collinearity of the rays is essential if the roots of the characteristic equation are distinct. In the case of equal roots of the characteristic equation this property will be true if and only if the rays do not belong to a straight line.