A novel fast direct solver using the recently proposed generalized equivalence integral equation (GEIE) is presented. By eliminating the line-of-sight between distant subdomains of convex geometries, the GEIE essentially reduces the problems' dimensionality, thus producing highly compressible impedance matrices. The compression is facilitated using a multilevel non-uniform grid (NG) scheme, tailored to the GEIE. The high compressibility and fast compression sum up to a fast direct solver. The solver is applied to the problem of scattering from an essentially circular cylinder, thus facilitating the construction of a modified Green's function that is needed in the GEIE formulation. For the 2-dimensional (2D) case, compression to O(1) unknowns is achieved.