Frey, M. Adelmann, A. JACoW Geneva, Switzerland This paper covers the topic of finding matched, i.e. stationary, distributions for coasting beams that takes into account higher order moments of the charge distribution. We present a general approach based on Lie-Algebra and Truncated-Power-Series Algebra, where the non-linear space-charge maps up to k-th order are constructed. The physical input parameters are intensity, projected emittances, and the energy. Our general approach is firstly compared and benchmarked with the well-known linear theory applied to the PSI Injector-2, the Ring Cyclotron and the IsoDAR machine. Secondly we compared the non-linear approach to PIC calculations. Open questions such as constraining the higher order moments will be discussed as well as applications beyond cyclotrons.