The pioneering work of Tartar and DiPerna established the existence of entropy solutions to systems of two conservation laws in the L^∞ setting by adapting the compensated compactness method. However, no regularity properties are known for these solutions. As a first step in this direction, we discuss a Liouville-type theorem for isentropic solutions of 2×2 genuinely nonlinear systems, together with some of its applications. The arguments are based on a kinetic formulation and on a Lagrangian representation at the kinetic level. Joint work with F. Ancona and E. Marconi.