A functional equation originating from elliptic curves

We obtain the general solution and the stability of the functional equation f(x+y+z,u+v+w)+f(x+y-z,u+v+w)+2f(x,u-w)+2f(y,v-w) =f(x+y,u+w)+f(x+y,v+w)+f(x+z, u+w)+f(x-z,u+v-w)+f(y+z,v+w)+f(y-z,u+v-w). The function f(x,y)= x³ +ax+b- y² having level curves as elliptic curves is a solution of the above functional equation.