Darboux Transformations for orthogonal differential systems and differential Galois Theory.

Darboux developed an algebraic mechanism to construct an infinite chain of "integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for $\sym^2 (\mathrm{SL}(2,\mathbb{C}))$ systems and, as a consequence, also for $\mathfrak{so}(3, C_K)$ systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials.