Let $H(x,p) + \alpha U(x)$ be a Hamiltonian system in $2n$-dimensional phase space with canonical coordinates $x_i,p_i$. Thus the Hamilton-Jacobi equation would take the form $H(x,p) + \alpha U(x) = E$. Assume that for every value of the parameter $\alpha$ the system admits a constant of the motion $K(\alpha)$ analytic in $\alpha$.

Coupling constant metamorphosis: The Hamiltonian $H'= \frac{H-E}{U}$ admits the constant of the motion $K' = K(-H')$, where now $E$ is a parameter.

So we can switch between coupling constant and energy levels and (super)integrability of the problem stays the same.