  A detailed study of a model for strongly-interacting fermions with exclusion
rules and lattice $\mathcal N=2$ supersymmetry is presented. A submanifold in
the space of parameters of the model where it is Bethe-ansatz solvable is
identified. The relation between this manifold and the existence of additional,
so-called dynamic, supersymmetries is discussed. The ground states are analysed
with the help of cohomology techniques, and their exact finite-size Bethe roots
are found. Moreover, through analytical and numerical studies it is argued that
the model provides a lattice version of the $\mathcal N=1$ super-sine-Gordon
model at a particular coupling where an additional $\mathcal N=(2,2)$
supersymmetry is present. The dynamic supersymmetry is shown to allow an exact
determination of the gap scaling function of the model.
