  The three-dimensional parallel spinor flow is the evolution flow defined by a
parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove
that, despite the fact that Lorentzian metrics admitting parallel spinors are
not necessarily Ricci flat, the parallel spinor flow preserves the vacuum
momentum and Hamiltonian constraints and therefore the Einstein and parallel
spinor flows coincide on common initial data. Using this result, we provide an
initial data characterization of parallel spinors on Ricci flat Lorentzian
four-manifolds, which in turn yields the first initial data characterization of
Ricci-flat pp-waves. Furthermore, we explicitly solve the left-invariant
parallel spinor flow on simply connected Lie groups, obtaining along the way
necessary and sufficient conditions for the flow to be immortal. These are, to
the best of our knowledge, the first non-trivial examples of evolution flows of
parallel spinors. Finally, we use some of these examples to construct families
of $\eta\,$-Einstein cosymplectic structures and to produce solutions to the
left-invariant Ricci flow in three dimensions. This suggests the intriguing
possibility of using first-order hyperbolic spinorial flows to construct
special solutions of curvature flows in Riemannian signature.
