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