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We establish deep and remarkable connections among partial differentialequations (PDEs) integrable by different methods: the inverse spectraltransform method, the method of characteristics and the Hopf-Coletransformation. More concretely, 1) we show that the integrability properties(Lax pair, infinitely-many commuting symmetries, large classes of analyticsolutions) of (2+1)-dimensional PDEs integrable by the Inverse ScatteringTransform method ($S$-integrable) can be generated by the integrabilityproperties of the (1+1)-dimensional matrix B\"urgers hierarchy, integrable bythe matrix Hopf-Cole transformation ($C$-integrable). 2) We show that theintegrability properties i) of $S$-integrable PDEs in (1+1)-dimensions, ii) ofthe multidimensional generalizations of the $GL(M,\CC)$ self-dual Yang Millsequations, and iii) of the multidimensional Calogero equations can be generatedby the integrability properties of a recently introduced multidimensionalmatrix equation solvable by the method of characteristics. To establish theabove links, we consider a block Frobenius matrix reduction of the relevantmatrix fields, leading to integrable chains of matrix equations for the blocksof such a Frobenius matrix, followed by a systematic elimination procedure ofsome of these blocks. The construction of large classes of solutions of thesoliton equations from solutions of the matrix B\"urgers hierarchy turns out tobe intimately related to the construction of solutions in Sato theory. 3) Wefinally show that suitable generalizations of the block Frobenius matrixreduction of the matrix B\"urgers hierarchy generates PDEs exhibitingintegrability properties in common with both $S$- and $C$- integrableequations.

On the remarkable relations among PDEs integrable by the inverse  spectral transform method, by the method of characteristics and by the  Hopf-Cole transformation