In this paper, we caracterize the solutions of specific bi-variate functional equations. The unknown functions represent the steady state distribution of specific two-dimensional random walks on Z2+. Inside the quarter plane, the jump have amplitude one, but are arbitrary on the axes. The main result is a necessary and sufficient condition for the solutions to be algebraic, when the "group" of the random walk (associated to an algebraic curve Q(x,y) = 0 having genus one) is finite. The method is based on Hilbert's factorization theorems, together with a uniformisation by means of elliptic functions.