For the \(J_2\) -problem, the mean-to-osculating transformation that guarantees the exact separation of long- and short-period terms up to the second order of \(J_2\) has been recently reported by the author. The transformation was computed in Delaunay canonical variables. However, because such kind of solution is derived from a vectorial generating function, it can be obtained in arbitrary variables—either singular or not, canonical or non-canonical—without need of recomputing the perturbation solution. The procedure is illustrated for the insightful set of semi-equinoctial variables. It could be equally applied, if desired, to compute a second-order, closed-form, analytical model truly consistent with the Draper semianalytical satellite theory.
