Liouville metric geodesic integral ($K=(U+V)(V{\dot{u}}^2-U{\dot{v}}^2)$)

For a surface metric $(U(u)+V(v))(d{u}^2+d{v}^2)$ with $U+V>0$, an affinely parametrized geodesic has constant energy $E=(U+V)({\dot{u}}^2+{\dot{v}}^2)/2$. Put $p=(U+V)\dot{u}$. The Euler-Lagrange equations give $\dot{p}=E{U}^{\prime }/(U+V)$. Thus $K=p^2-2EU$ has derivative zero, proving the displayed additional quadratic integral. This separation structure is a geometric example of Liouville integrability.