Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cyclesâ€”which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).

However, $H(x,y) = y/x^2 = c$ is not preserved at $x = 0$, so it is not integrable, even though it has an explicit solution.