Differential equation in the torus

[rmiranda]

Hello all.
Consider the torus $$T^2$$ as a subset of R^3, for example the inverse image of 0 by the function $$f(x,y,z)=(\sqrt{x^2+y^2}-1)^2+z^2-4$$.
I need to obtain a example of a vector field $$X$$ defined in the whole $$R^3$$, such that:
1) $$X$$ is invariant in the torus
2) the orbits of $$X$$ in the torus are all periodic of the same period (I thought in something like the orbits being the parallels).

I can obtain such a v.f. in cylindrical coordinates, but when I put my example in cartesian coords, the equations are turning to be very complicated to my purpose, may be someone has a simpler example of such v.f.?

 

[CompuChip]

I can obtain such a v.f. in cylindrical coordinates

So is that a problem? Do you have to give it in a specific coordinate system? If you can show that the vector field is defined on all of space and satisfies the requirements in cylindrical coordinates, the argument is just as valid as in Cartesian ones, isn't it?