Finiteness of couples of singular modular invariants on a non-modular two-dimensional algebraic curve

Let S be an irreducible algebraic curve in the affine complex plane. Assume that S is neither a horizontal line, nor a vertical line, nor a modular cu rve Y-0(N) (for any integer N greater than or equal to 1). Then there are o nly finitely many points P of S such that both coordinates of P are singula r moduli (i.e. invariants of elliptic curves with complex multiplication).