An analytical description of neutrino behavior in external matter and fields and specifically that of neutrino spin dynamics not only provides the understanding of the basics of neutrino spin oscillations in astrophysical media but also enables to discover and give account for another new phenomena. Examples include the effects of neutrino Spin Light \cite{StudTern_PLB2005}, and neutrino beam self-polarization in matter \cite{StudTern_PLB2005,LobStud_PLB2004}. All these effects arise due to helicity dependence of neutrino energy in non-moving matter. In this study we go further in developing the neutrino theory in a medium and consider matter motion. In this case the helicity is no longer the exact quantum number and for consistent treatment one has to find the spin integral of motion. Once found, it enables easy derivation of the neutrino dispersion relation and clear physical interpretation of the spin quantum number entered in it.