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

Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions