High brightness beam transport in a non-linear focusing channel, while avoiding beam emittance growth and halo formation, is considered. Invariance of beam distribution function is treated as a problem of proper matching of the beam with focusing channel. Matched conditions for intense beam are obtained from the stationary Vlasov’s equation for beam distribution function and Poisson’s equation for electrostatic beam potential. Two approaches to self-consistent problem for matched beam distribution are treated: (i) focusing field is determined by given beam distribution, and (ii) matched beam distribution is defined by focusing potential of the structure. Attained solutions provide theoretical basis for choosing parameters of the space charge dominated beam transport with suppressed halo. ©1998 American Institute of Physics.