RSolve[{H1[x0 + t (-y0 + x0 y0^2 + x0^3),
y0 + t (x0 + x0^2 y0 + y0^3)] ==
H1[x0, y0] + t (H1[x0, y0] - H2[x0, y0]),
H2[x0 + t (-y0 + x0 y0^2 + x0^3), y0 + t (x0 + x0^2 y0 + y0^3)] ==
H2[x0, y0] + t (H1[x0, y0] + H2[x0, y0])}, {H1, H2}, {x0, y0}]