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We propose a new approach that allows one to reduce nonlinear equations onLie groups to equations with a fewer number of independent variables forfinding particular solutions of the nonlinear equations. The main idea is toapply the method of noncommutative integration to the linear part of anonlinear equation, which allows one to find bases in the space of solutions oflinear partial differential equations with a set of noncommuting symmetryoperators. The approach is implemented for the generalized nonlinearSchr\"{o}dinger equation on a Lie group in curved space with local cubicnonlinearity. General formalism is illustrated by the example of noncommutativereduction of the nonstationary nonlinear Schr\"{o}dinger equation on the motiongroup $E(2)$ of the two-dimensional plane $\mathbb{R}^{2}$. In the particularcase, we come to the usual ($1+1$) dimensional nonlinear Schr\"{o}dingerequation with the soliton solution. Another example provides the noncommutativereduction of the stationary multidimensional nonlinear Schr\"{o}dinger equationon the four-dimensional exponential solvable group.

Noncommutative reduction of the nonlinear Schr\"{o}dinger equation on Lie groups