Solutions of the loop equations of a class of generalized Frobenius manifolds

We prove the existence and uniqueness of solution of the loop equation associated with a semisimple generalized Frobenius manifold with non-flat unity, and show, for a particular example of one dimensional generalized Frobenius manifold, that the deformation of the Principal Hierarchy induced by the solution of the loop equation is the extended q-deformed KdV hierarchy.