On Lax pairs and matrix extended simple Toda systems

Noncommutative theories have been studied and probed from different viewpoints (see reviews [18, 34, 48]). For instance, a number of noncommutative generalizations of integrable systems were presented (see, e.g., [9, 16, 17, 24, 39]). Solutions were investigated using the dressing method and Riemann-Hilbert problems, formulations, and properties such as infinite sets of conserved quantities were shown, and linear systems (or Lax pairs) were exhibited in different articles (e.g., [16, 17, 23, 30, 39, 40, 41]). Systems such as solitons, instantons, monopoles, Yang-Mills-Higgs, and nonlinear sigma models in 2+ 1 dimensions have been explored with noncommutative variables, raising certain similarities (see, e.g., [22, 29]). Noncommutative (Yang-Mills) theories have been linked to string theories with nontrivial B-field. It is known that self-dual Yang-Mills equations in 4-dimensional space (or their generalizations) lead, through reductions, to many integrable systems in lower dimensions, and for this, they have also been initially labelled as “master equations” [52]. Similarly, supersymmetric integrable systems in dimensions smaller than 4 have also been found to have a reduction relation with respect to supersymmetric self-dual Yang-Mills equations. As a first step towards a (possible) noncommutative “master system,” it has then been mentioned that a noncommutative version of self-dual supersymmetric YangMills systems could provide via reductions noncommutative generalizations of (supersymmetric) integrable systems [35]. In the following, our general interest is twofold: noncommutativity and deformations. Noncommutativity can be introduced using different structures, as shown in [33]. A basic noncommutativity of variables could be imposed through [xμ,xν] = iθμν, and can be associated to a ∗-product. Use of these noncommutative variables (or ∗-product) could also be seen as probing deformations of (integrable) systems, with a deformation