It was shown recently that Frobenius reduction of the matrix fields reveals
interesting 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 of
characteristics (Ch-integrable PDEs). However, only two classes of
S-integrable PDEs have been involved: soliton equations like Korteweg-de
Vries, Nonlinear Shr\"odinger, Kadomtsev-Petviashvili and Davey-Stewartson
equations, and GL(N,\CC) Self-dual type PDEs, like Yang-Mills equation. In
this paper we consider the simple five-dimensional nonlinear PDE from another
class of S-integrable PDEs, namely, scalar nonlinear PDE which is
commutativity condition of the pair of vector fields. We show its origin from
the (1+1)-dimensional hierarchy of Ch-integrable PDEs after certain
composition of Frobenius type and differential reductions imposed on the matrix
fields. Matrix generalization of the above scalar nonlinear PDE will be derived
as well.Comment: 14 pages, 1 figur