  A generalisation of the non--perturbatively stable solutions of string
equations which respect the KdV flows, obtained recently for the $(2m-1,2)$
conformal minimal models coupled to two--dimensional quantum gravity, is
presented for the $(p,q)$ models. These string equations are the most general
string equations compatible with the $q$--th generalised KdV flows. They
exhibit a close relationship with the bi-hamiltonian structure in these
hierarchies. The Ising model is studied as a particular example, for which a
real non-singular numerical solution to the string susceptibility is presented.
