Let $G=S^1$ and $X=S^3\times D^4\subset \Bbb C^2\times \Bbb C^2=\Bbb C^4$. Two $G$-actions on $X$ are said to be equivalent if there is an orientation-preserving $G$-equivariant diffeomorphism $X\to X$. In the second page of this paper (https://www.jstor.org/stable/60608), it is written that: up to equivalence, there are only two distinct free $G$-actions on $X$. The first one is given by $z\cdot (u_1,u_2,v_1,v_2)=(zu_1,zu_2,v_1,v_2)$, and the second one is given by $z\cdot (u_1,u_2,v_1,v_2)=(zu_1,zu_2,zv_1,v_2)$.
It seems that there is no proof given in the paper. How can this be proved?