  A complete classification of integrable conservative hydrodynamic chains is
presented. These hydrodynamic chains are written via special coordinates --
moments, such that right hand sides of these infinite component systems depend
linearly on a discrete independent variable $k$. All variable coefficients of
these hydrodynamic chains can be expressed via modular forms with respect to
moment $A^{0}$, via hypergeometric functions with respect to moment $A^{1}$;
they depend polynomially on moment $A^{2}$ and linearly on all other higher
moments $A^{k}$. A dispersionless Lax representation is found. Corresponding
collisionless Boltzmann (Vlasov like kinetic) equation is derived. A Riemann
mapping is constructed. A generating function of conservation laws and
commuting flows is presented.
