The main series of nonhomogeneous systems of hydrodynamic type with constant matrices possessing $1^{st}$ order conservation laws

Let $u_t^i=v_iu_x^i+f_i(u)$, $i=1,\ldots,n$ be a system of PDE with constant $v_i$. We give a classification of such systems possessing nontrivial conservation laws $dg(u,u_x)/dt=dh(u,u_x)/dx$ and their explicit form for two main series. The third (and the last) main series is shown to be nontrivial. Some examples of such systems are new.