Wahba's problem is the following:

$$\min_R \sum_{k=1}^K \|v_k - Rw_k\|^2$$
where $v_k$ and $w_k$ are arbitrary $3\times 1$ vectors, and $R$ is a rotation matrix (i.e., orthogonal with $\det(R)=1$).
A variant of Wahba's problem is the orthogonal Procrustes problem.

I have a more complicated variant of Wahba's problem. 
- I only study $K=3$.
- I have $w_k=R_{u_k}(\alpha_k)v_k$ where $R_{u_k}(\alpha_k)$ is a rotation around the axis $u_k$ with an angle $\alpha_k$.
- The axes $u_k$ are given in advance.
- I can optimize over $\alpha_k$.

Thus, my variant of Wahba's read
$$J_{\min}=\min_{R, \alpha_k} \sum_{k=1}^3 \|v_k - RR_{u_k}(\alpha_k)v_k\|^2$$
where $v_k$ and $u_k$ are given. My claim is that there exist non-trivial solutions, i.e., $R\neq I$ and $\alpha_k\neq 0$ so that the minimum value is $J_{\min}=0$. Note that for $R=I, \alpha_k=0$, we trivially obtain $J_{\min}=0$. To be more precise, what I observed in matlab is that if I choose $v_k$ and $u_k$ randomly, I always have multiple solutions achieving $J_{\min}=0$.



Based on the (partial) answer below, I generated the following figure. I found 4 solutions. I plot for a special case here where two circles are parallel, and the third perpendicular to the other two.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/rtY7i.png