  There exist many examples of systems which have some symmetries, and which
one may monitor with symmetry preserving controls. Since symmetries are
preserved along the evolution, full controllability is not possible, and
controllability has to be considered inside sets of states with same
symmetries. We prove that generic systems with symmetries are controllable in
this sense. This result has several applications, for instance: (i) generic
controllability of particle systems when the kernel of interaction between
particles plays the role of a mean-field control; (ii) generic controllability
for families of vector fields on manifolds with boundary; (iii) universal
interpolation for neural networks architectures with "generic" self
attention-type layers - a type of layers ubiquitous in recent neural networks
architectures, e.g., in the Transformers architecture. The tools we develop
could help address various other questions of control of equivariant systems.
