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It was shown recently that Frobenius reduction of the matrix fields revealsinteresting relations among the nonlinear Partial Differential Equations (PDEs)integrable by the Inverse Spectral Transform Method ($S$-integrable PDEs),linearizable by the  Hoph-Cole substitution ($C$-integrable PDEs) and integrable by the method ofcharacteristics ($Ch$-integrable PDEs). However, only two classes of$S$-integrable PDEs have been involved: soliton equations like Korteweg-deVries, Nonlinear Shr\"odinger, Kadomtsev-Petviashvili and Davey-Stewartsonequations, and $GL(N,\CC)$ Self-dual type PDEs, like Yang-Mills equation. Inthis paper we consider the simple five-dimensional nonlinear PDE from anotherclass of $S$-integrable PDEs, namely, scalar nonlinear PDE which iscommutativity condition of the pair of vector fields. We show its origin fromthe (1+1)-dimensional hierarchy of $Ch$-integrable PDEs after certaincomposition of Frobenius type and differential reductions imposed on the matrixfields. Matrix generalization of the above scalar nonlinear PDE will be derivedas well.

On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields