Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose boundary 2-sphere corresponds to the singular isotropy group (stabilizer) $K=SO(2)$ and interior corresponds to the principal isotropy group $H=id$. The problem is to classify all such 6 manifolds admitting such actions.

In Bredon's book "Introduction to Compact Tranformation Groups", he has a way to classify group actions with 2 orbit types, which was initially due to Janich. According to Corollary 6.2 on page 257, Chapter V.6 of the book, the class of such 6-manifolds is in bijection with the set $[S^2,N(H)/(N(H)\cap N(K))]/\pi_0(N(H)/H)=\pi_2(N(H)/(N(H)\cap N(K)))/\pi_0(N(H)/H)=\pi_2(\mathbb{RP^2})=\mathbb{Z}.$

Here $K=SO(2),\ H=id$ are the isotropy groups (stabilizers) given in the first paragraph, and $N(H)=id,\ N(K)=O(2)$ are their normalizers in $G=SO(3)$. **Thus such actions are parametrized by integers.**

But Bredon's construction in the book is rather involved. I'm aiming at simple explicit description of this family of $SO(3)$-actions. Moreover, I'm wondering **which of those manifolds are simply connected**. I'm mainly interested in simply connected examples.

Two more remarks:

Remark 1. If we ask the orbit space to be a 2-disk instead of 3-ball, then in Bredon's book he gave explicit classification of such $G$-spaces. More precisely, all $O(n)$-spaces with isotropy groups $O(n-1)$ or $O(n-2)$ and orbit space $D^2$ are given by $\Sigma_k^{2n-1}=S^{n-1}\times D^n \cup_{\varphi^k} S^{n-1}\times D^n$, where $\varphi$ is an $O(n)$-equivariant gluing map. For details see Chapter I.7 and V.6 of Bredon's book "Introduction to Compact Tranformation Groups". Those spaces have alternative description: the Brieskorn varieties $B^{2n-1}_k=\{z_0^k+z_1^2+\cdots z_n^2=0\}\cap S^{2n+1}$, see Chapter V.9 of the book. My goal is to find such explicit description in the case when the orbit space is a 3-ball and the group $G=SO(3)$.

Remark 2. I have one example of such actions. Consider the 4-dimensional complex representation $\mathbb{C}^2\oplus \mathbb{C}^2$ of $SU(2)$, take its projectivization, then we get a linear $SU(2)$-action on $\mathbb{CP}^3$, which is ineffective (not faithful) and descends to an effective $SO(3)$-action. This action satisfies the requirement. And my intuition tells me that this action corresponds to the integer parameter 0. But I don't know how to show it.

I'd like to point out that this question is related to another question I asked: SO(3) action on (simply connected) 6 manifold with discrete fixed point. They are both about $SO(3)$-actions on 6-manifolds, but under different conditions.

Update: I suspect they are $S^2$ bundles over $S^4$, as I construct such actions on $\mathbb{CP}^3$ and $S^2\times S^4$.