  Similarly to the complex matrix model, the rainbow tensor models are
superintegrable in the sense that arbitrary Gaussian correlators are explicitly
expressed through the Clebsh-Gordan coefficients. We introduce associated
(Ooguri-Vafa type) partition functions and describe their $W$-representations.
We also discuss their integrability properties, which can be further improved
by better adjusting the way the partition function is defined. This is a new
avatar of the old unresolved problem with non-Abelian integrability concerning
a clever choice of the partition function. This is a part of the long-standing
problem to define a non-Abelian lift of integrability from the fundamental to
generic representation families of arbitrary Lie algebras.
