Extended discrete kinetic theory including sources, sinks, creation and absorption of test particles, inelastic scattering,… added to the elastic collisions, was introduced by Boffi and Spiga. For the mass conservation law (or momentum, energy), polynomials of the mass (or densities) are added, leading to a lack of conservation equations. We consider models with linear nonconservative terms, LNC (Spiga- Platkowski) and quadratic, QNC (Piechor-Platkowski). For quasi-linear systems of PDE (linear differential terms and quadratic nonlinearities), we present a general formalism for the determination of stationary, similarity, periodic and (1 + 1) dimensional exact solutions in one space variable. We present results for the two-velocity, the two and three dimensional Broadwell, the hexagonal 6viand the two-squares 8viDVMs (Discrete Velocity Models). The similarity solutions are obtained from the compatibility between different scalar nonlinear Riccati equations (NLODE), Firstly, we apply this method to the similarity solutions of the QNC models, while for (1 + 1) solutions only one density is not constant.