How to make an object move on a Mobius strip? I have Mobius strip and a sphere. I want the sphere to move on the strips surface, like on a path. How can I do this?

I've tried drawing curves, but I can't create the whole path at once and I've tried joining drawed pieces but the sphere moves just on the first drawed curve. I've tried joining two circles but also the sphere goes just on one of them. There is a way to create a mobius nod directly but it doesn't match the shape and I can't make it match. Can I somehow draw the path by code? 
I'm very new to this so if there's an easier way I would appreciate it. 

Tryed making edges the path, didn't work.
Create a Bezier Curve. Open it (it shouldn't be closed). Tilt one end 180°. &#9096; CtrlT
The Curve should resemble the twist of the moebius strip.

Add a Follow Path Constraint to the object. Check Fixed Position and set a key with the offset at 0 and 1.

Select the created keyframes and make them linear. T > L
Then make the graph cyclic. &#8679; ShiftE > Make Cyclic  
Set a key on the first frame of the rotation. I
Set a key after the second keyframe of the offset on the rotation with the object rotated 180°, making it ready for the second circulation.
Set the new keys to constant interpolation. T > C



Wheeee!