Matrix extension of multidimensional dispersionless integrable hierarchies

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