  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.
