  In this paper we prove that the shift map defined on the moduli space of
twisted pentagram spirals of type $(N, 1)$ possesses a non-standard Lax
representation with an associated monodromy whose conjugation class is
preserved by the map. We prove this by finding a coordinate system in the
moduli space of twisted spirals, writing the map in terms of the coordinates
and associating a natural parameter-free non-standard Lax representation. We
then show that the map is invariant under the action of a $1$-parameter group
on the moduli space of twisted $(N,1)$ spirals, which allows us to construct
the Lax pair. We also show that the monodromy defines an associated Riemann
surface that is preserved by the map. We use this fact to generate invariants
of the shift map.
