  We apply Noether's theorem to observations of main-sequence stars from the
Gaia Data Release 2 archive to probe the matter distribution function of the
Galaxy. That is, we examine the axial symmetry of stars at vertical heights
$z$, $0.2 \le |z| \le 3$ kpc, to probe the quality of the angular momentum
$L_z$ as an integral of motion. The failure of this symmetry test would speak
to a Milky Way, in both its visible and dark matter, that is not isolated
and/or not in steady state. The left-right symmetry-breaking pattern we have
observed, north and south, reveals both effects, with a measured deviation from
symmetry of typically 0.5%. We show that a prolate form of the gravitational
distortion of the Milky Way by the Large Magellanic Cloud, determined from fits
to the Orphan stream by Erkal et al., 2019, is compatible with the size and
sign of the axial-symmetry-breaking effects we have discovered in our sample of
up to 14.4 million main-sequence stars, speaking to a distortion of an
emergent, rather than static, nature.
