  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 B\"urgers 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 $GL(M,\CC)$ 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 B\"urgers hierarchy turns out to
be intimately related to the construction of solutions in Sato theory. 3) We
finally show that suitable generalizations of the block Frobenius matrix
reduction of the matrix B\"urgers hierarchy generates PDEs exhibiting
integrability properties in common with both $S$- and $C$- integrable
equations.
