Dirac's quantization of the (2+1)-dimensional analog of Ashtekar's approach to quantum gravity is investigated. After providing a diffeomorphism-invariant regularization of the Hamiltonian constraint, we find a set of solutions to this Hamiltonian constraint which is a generalization of the solution discovered by Jacobson and Smolin. These solutions are given by particular linear combinations of the spin network states. While the classical counterparts of these solutions have a degenerate metric, due to a ‘quantum effect’, the area operator has a non-vanishing action on these states. For computational simplicity, we restricted the analysis to piecewise analytic graphs with four-point vertices and to finite-dimensional representations of SL(2, R) . It is considered, however, that the analysis will have to be extended to more generic cases in order to obtain geometrodynamical states. We also discuss how to extend our results to (3+1) dimensions.