'Determine 2 DOF 3D rotation knowing a single vector

I am looking for some good idea to solve the following problem.

These elements are known:

a unit vector v is seen in cartesian coordinates in two reference frames A and B as va and vb.
frame B is obtained with a 2 DOF unknown rotation of frame A of the form q (quaternion) = qx * qz; where qx = {cos(rx/2),sin(rx/2),0,0} and qz = {cos(rz/2),0,0,sin(rz/2)}.

Problem: determine rx and rz angles.

Thanks for any reply.

3d rotation quaternions

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