  Superintegrable models are very special dynamical systems: they possess more
conservation laws than what is necessary for complete integrability. This
severely constrains their dynamical processes, and it often leads to their
exact solvability, even in non-equilibrium situations. In this paper we
consider special Hamiltonian deformations of superintegrable quantum circuits.
The deformations break superintegrability, but they preserve integrability. We
focus on a selection of concrete models and show that for each model there is
an (at least) one parameter family of integrable deformations. Our most
interesting example is the so-called Rule54 model. We show that the model is
compatible with a one parameter family of Yang-Baxter integrable spin chains
with six-site interaction. Therefore, the Rule54 model does not have a unique
integrability structure, instead it lies at the intersection of a family of
quantum integrable models.
