This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is still an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I have also established the explicit expression of general solutions on the torus and on the real line in my work [hal-04227081].