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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 $(H,L,P)$.

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    @Qmechanic I just clarified the Lagrangian I'm dealing with. – 2017-02-14
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    @Qmechanic Are the initial conditions not sufficient to describe the dynamics? This is what I use to perform simulations. – 2017-02-14
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    There's another type of transformation that conserves all three quantities: time evolution. If $\{x_i(0),\dot{x}_i(0)\}_{i=1}^n$ is a set of initial conditions with a particular $H$, $P$, and $L$, then so is $\{x'_i(0),\dot{x}'_i(0)\} = \{x_i(t),\dot{x}_i(t)\}$ for any value of $t$. – 2017-02-14
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    @MichaelSeifert That's a good point. – 2017-02-14

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My first thought is that transformations from the non-relativistic Poincare group, the Galilean transformations, conserve $H$, $P$, and $L$.

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    Unless I'm interpreting you wrong, that's not true. If $\vec{P} = 0$ in one reference frame, it will generally not be zero in another frame. – 2017-02-14