  The L\'evy-Leblond-Newton (LLN) equation for non-relativistic fermions with a
gravitational self-interaction is reformulated within the framework of a
Bargmann structure over a $(n+1)$-dimensional Newton-Cartan (NC) spacetime. The
Schr\"odinger-Newton (SN) group, introduced in [21] as the maximal group of
invariance of the SN equation, turns out to be also the group of conformal
Bargmann automorphisms preserving the coupled L\'evy-Leblond and NC
gravitational field equations. Within the Bargmann geometry a generalization of
the LLN equation is provided as well. The canonical projective unitary
representation of the SN group on 4-component spinors is also presented. In
particular, when restricted to dilations, the value of the dynamical exponent
$z=(n+2)/3$ is recovered as previously derived in [21] for the SN equation.
Subsequently, conserved quantities associated to the (generalized) LLN equation
are also exhibited.
