Within the context of the gravitational n-body problem I'm interested in perturbations of the initial conditions $\{x_i(0),\dot{x}_i(0)\}_{i=1}^n$ which leave all the constants of motion unchanged.
It's clear to me that the linear momentum($P$) is preserved under any transformation of the initial position vectors and energy($H$) is preserved under any isometry applied to initial position vectors. However, when I add the constraint of conserving angular momentum($L$), the general nature of these transformations which would leave $(H,L,P)$ unchanged is not clear.
Note: Michael Seifert made a good point that time evolutions would yield new initial conditions that preserve $(H,L,P)$.
My first thought is that transformations from the non-relativistic Poincare group, the Galilean transformations, conserve $H$, $P$, and $L$.