Summary: A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called -shock wave, where is th derivative of the Dirac delta function . In this paper the case is studied in details. We introduce a definition of -shock wave type solution for the system
Within the framework of this definition, the Rankine-Hugoniot conditions for -shock are derived and analyzed from geometrical point of view. We prove -shock balance relations connected with area transportation. Finally, a solitary -shock wave type solution to the Cauchy problem of the system of conservation laws , , with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of -shocks) but their derivatives as well.