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We represent an algorithm allowing one to construct new classes of partiallyintegrable multidimensional nonlinear partial differential equations (PDEs)starting with the special type of solutions to the (1+1)-dimensional hierarchyof nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (theB\"urgers hierarchy).  We derive examples of four-dimensional nonlinear matrix PDEs together withthey scalar and three-dimensional reductions. Variants of theKadomtsev-Petviashvili type and Korteweg-de Vries type equations arerepresented among them. Our algorithm is based on the combination of twoFrobenius type reductions and special differential reduction imposed on thematrix fields of integrable PDEs. It is shown that the derived four-dimensionalnonlinear PDEs admit arbitrary functions of two variables in their solutionspaces which clarifies the integrability degree of these PDEs.

Differential reductions of the Kadomtsev-Petviashvili equation and associated higher dimensional nonlinear PDEs