It is well known that the only principal *locally trivial* fiber bundle $S^1 \to S^3 \to S^2$ is Hopf map $h$ (see, for example, [1]).

What if we drop the *local triviality* but mantain a *"principality"* hypothesis meaning that the fibers are given by an $S^1$ action? Are there well known families of such bundles $S^1 \to S^3 \to S^2$ whose homotopy class is not that of $h$?

In particular, are the homotopy class of the nontrivial multiples $k[h]$ of Hopf's map in $\pi_3(S^2)$, $k \not\in \{ \pm 1,0 \}$, given by such bundles?

Such bundles would be necessarily *not* locally trivial, but hopefully they can be fibrations.