Matrix extension of multidimensional dispersionless integrable hierarchies

We consistently develop a recently proposed scheme of matrix extensions of dispersionless integrable systems in the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax–Sato equations, matrix equations on the background of vector fields, and the dressing scheme. Reductions, the construction of solutions, and connections to geometry are discussed. We separately consider the case of an Abelian extension, for which the Riemann–Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of the Penrose formula in curved space.