A generalization of addition formulae

Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X J, M : J R and H : J ² R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.