  The coupling of $N=1$ SCFT of type $(4m,2)$ to two-dimensional supergravity
can be formulated non-perturbatively in terms of a discrete super-eigenvalue
model proposed by Alvarez-Gaum\'e, et al. We derive the superloop equations
that describe, in the double scaling limit, the non-perturbative solution of
this model. These equations are equivalent to the double scaled super-Virasoro
constraints satisfied by the partition function. They are formulated in terms
of a $\widehat c=1$ theory, with a $\IZ_2$-twisted scalar field and a
Weyl-Majorana fermion in the Ramond sector. We have solved the superloop
equations to all orders in the genus expansion and obtained the explicit
expressions for the correlation functions of gravitationally dressed scaling
operators in the NS- and R-sector. In the double scaling limit, we obtain a
formulation of the model in terms of a new supersymmetric extension of the KdV
hierarchy.
