We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf–Cole transformation. More concretely, (1) we show that the integrability properties (Lax pair, infinitely-many commuting symmetries, large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the inverse scattering transform method (S-integrable) can be generated by the integrability properties of the (1+1)-dimensional matrix Bürgers hierarchy, integrable by the matrix Hopf–Cole transformation (C-integrable). (2) We show that the integrability properties (i) of S-integrable PDEs in (1+1) dimensions, (ii) of the multidimensional generalizations of the self-dual Yang–Mills equations and (iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced multidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius matrix, followed by a systematic elimination procedure of some of these blocks. The construction of large classes of solutions of the soliton equations from solutions of the matrix Bürgers hierarchy turns out to be intimately related to the construction of solutions in Sato theory. (3) Finally, we show that suitable generalizations of the block Frobenius matrix reduction of the matrix Bürgers hierarchy generates PDEs exhibiting integrability properties in common with both S- and C-integrable equations.