Difference equation for functions analytical outside two squares

We consider a totality of two squares constructed by primitive periods $1$ and $i$ and sufficiently close to each other. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions analytical outside this set and vanishing at infinity. We indicate applications to the moments problem for entire functions of exponential type.