On the Nonlinear Operators Having Jacoby Matrix Commuting with a Ring of the Constant Matrix

The criterion of existence of nonlinear operator $\mathbf{u}:C^m\rightarrow C^m \left(m\ge2\right)$ for which it's Jacoby matrix commutes with every constant complex matrix from any given ring $Q$ is obtained. The main theorem says that such operator exists if and only if a ring $Q$ has at least one $\left(r,l\right)$-pair.