Do you mean a particle is restricted to move on a torus surface (inside a torus and always in contact with the surface)?
The particle is always in contact with the surface, ie never change toroid.
What is the interaction force between particle and the torus surface? Normal force ?
Only normal force
or Is there another force to keep particle always in contact with the torus surface (it might fall down due to gravity).

Assume the larger radius is R, and smaller radius is r for the torus.
Does $z=r \sin\beta$, and $\tan\gamma=y/x$ in your case?
Yes
I do not understand the meaning of $\alpha$ in the attached picture.
The largest radio is called $c$ and the radius of the ring is not called $\alpha$, this is called $a$.
Your problem might be similar to a circular motion in vertical direction , plus a constant angular speed rotational motion (angular momentum is conserved).
If something similar. The treatment of the problem has been with the Hamilton's mechanics. I can send a pdf file, the bad is that this in Spanish.
Thanks from Bogotá, Colombia
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