I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image $S^2$ more precisely:

The Hopf fibration $p:S^3\to S^2$ is defined by $$(z_1,z_2)\mapsto (2z_1\overline{z_2},|z_1|^2-|z_2|^2)$$ I am trying to construct an n parameter family of measure preserving real analytic diffeomorphisms $\{f_{(\theta_1,\ldots,\theta_n)}\}_{(\theta_1,\ldots\theta_n)\in T^n}$ $$f_{(\theta_1,\ldots,\theta_n)}:S^3\to S^3$$ Satisfying the following two properties:

1> Given any $x,y\in S^3$, there exists $(\theta_1,\ldots,\theta_n)\in T^n$ such that $p(f_{(\theta_1,\ldots,\theta_n)}(x))=p(y)$.

2> Given any $x,y\in S^3$ satisfying $p(x)=p(y)$, then $f_{(\theta_1,\ldots,\theta_n)}(x)=f_{(\theta_1,\ldots,\theta_n)}(y) \;\;\forall\; (\theta_1,\ldots,\theta_n)\in T^n$

I know how to do it if the second condition is removed. Not sure if it could be done with both conditions intact. $n$ can be any natural number. Smaller the better.