Non-standard couples of angles of rotations

Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. We show that on M, the smooth closure of the smooth volume-preserving conjugation class of some Liouville rotations of angle alpha contains a smooth volume-preserving diffeomorphism T that is metrically isomorphic to an irrational rotation of angle beta on the circle, with alpha different of beta, and with alpha and beta chosen either rationally dependent or rationally independent. In particular, if M is the closed annulus, M admits a smooth ergodic pseudo-rotation T of angle alpha that is metrically isomorphic to the rotation of angle beta. Moreover, T is smoothly tangent to the rotation of angle alpha on the boundary of M