Solution is attained in approximation of high brightness beam.
Distribution function in phase space is determined as a stationary function of the energy integral. It is shown that particle distribution of the bright beam always tends to such a shape that the space charge beam potential is opposite to the external potential regardless of the applied field. Equipartitioning for beam distribution between degrees of freedom follows directly from the choice of stationary distribution function. Analytical expressions for r-z equilibrium beam profile and beam current limit in RF field are obtained.