Regularisation by Hamiltonian extension

We consider the Kepler potential and its relatives $q\mapsto -\mathrm{\parallel }q{\mathrm{\parallel }}^{-2(1-1\mathrm{/}n)}$, $n\in \mathbb{N}$ in arbitrary dimension $d$. We derive a unique real-analytic symplectic extension of phase space on which the Hamiltonian flow is complete and still real-analytic.