= Axisymmetric Stäckel third integral
{title2=$I$}
For $H=\tfrac12\sum p_j^2/h_j^2-\psi$, $l=p_\phi$ 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 $I=\tfrac12[\mu v_\lambda^2+\lambda v_\mu^2+(\lambda+\mu+\alpha)v_\phi^2]-[\mu G(\lambda)-\lambda G(\mu)]/(\lambda-\mu)$. Physical components are $v_j=h_j\dot q_j$ and $p_j=h_jv_j$. Separation proves conservation locally; direct <Poisson brackets> extend it across regular coordinate patches.
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