Axisymmetric Stäckel third integral ($I$)

For $H=\frac{1}{2}\sum p_j^2/h_j^2-\psi$, $l=p_{\varphi }$ and a Stäckel potential, multiplying the separated Hamilton-Jacobi equation by $\lambda -\mu$ gives $2(\tau +\alpha )(\tau +\beta )p_{\tau }^2-(\alpha -\beta )l^2/[2(\tau +\alpha )]-G(\tau )-E\tau =-I$ for both $\tau =\lambda ,\mu$. Eliminating $E$ gives $I=\frac{1}{2}[\mu v_{\lambda }^2+\lambda v_{\mu }^2+(\lambda +\mu +\alpha )v_{\varphi }^2]-[\mu G(\lambda )-\lambda G(\mu )]/(\lambda -\mu )$. Physical components are $v_j=h_j{\dot{q}}_j$ and $p_j=h_j{v}_j$. Separation proves conservation locally; direct Poisson brackets extend it across regular coordinate patches.