The detailed photometrical monitoring of the cataclysmic variable SDSS J080434.20+ 510349.2 was started at the Crimean astrophysical observatory (Ukraine) and Apache Point Observatory (USA) before the 2006 outburst, continued during the outburst and during two years after it. We established an unique nature of the primary and secondary components of the binary. We performed a comprehensive study of the white dwarf's pulsations recently discovered over five months after two years after the 2006 outburst. It is shown that the most stable pulsations are equal to the period of 12.6 min or it's twice value. On the basis of all the available observations, more precise values for orbital and superhump periods were found to be 0.0590048(3) days and 0.059729(4) days, respectively. Our estimation of the mass of secondary component lies in the range from 0.037 to 0.087 of solar mass. This confirms the previous suggestion that the secondary component is a brown dwarf with a high probability.