Topology of equipotential surfaces Let's think of planar equipotential surfaces, say they are parallel to the x-y plane, then apparently $p_x$ and $p_y$ are conserved quantities.
Next, let's move on to cylindrical equi-potentials. Then $p_z$ and $p_\phi$ are conserved from symmetries. The other way we can think about it is that a cylinder can be obtained from identifying two opposite borders of a rectangle. So we can map $p_x \rightarrow p_z, p_y\rightarrow p_\phi$.
Let's do the folding once more, so that the cylinder now becomes a torus. If you believe topology, then we should expect two conserved momenta, associated with orbiting the red circle in the image and the magenta circle. But careful calculation invoking Noether's theorem doesn't seem to support this belief because transformation along the red circle doesn't seem to preserved the Lagrangian and hence doesn't provide a conserved momentum. 
To expand a little more, I use the following coordinates on the torus, where $\theta$ is the angle on minor circles, and $\phi$ is on the major circle. Then the Lagrangian expressed in terms of $\theta$ and $\phi$ is dependent on $\theta$, so $p_\theta$ is not conserved. 
So, if you believe in topology, where is the other symmetry?
\begin{align}x(\theta,\varphi) &= (R+ r\cos\theta)\cos\varphi\\ y(\theta,\varphi) &= (R+ r\cos\theta)\sin\varphi\\ z(\theta,\varphi)&= r\sin\theta\end{align}

 A: Your represented (in the nice figure!) torus is non-flat and is immersed in $\mathbb R^3$, the one obtained by the identifications of the opposite edges of a rectangle is instead  flat and is not metrically immersed in $\mathbb R^3$.
These two kinds of torus  are topologically identical (homeomorphic and also diffeomorphic actually), but they are metrically distinct (they are not isometric). 
Here metrical notions matter. This is the reason why the immersed torus has one symmetry less than the flat torus.
The orbits of the angle $\theta$ are symmetries provided the metric on the torus is flat as it arises putting  the metric of the plane on the torus with the standard identifications just to produce a flat torus. However, this metric is not the one the torus receives from the metric of $\mathbb R^3$ viewing it as an immersed surface: the curvature shows up here and $\theta$ is not a metrically invariant direction. 
In the flat torus, for instance, all the violet circles  have the same length, in the immersed torus, their length is variable depending on $\theta$ as it is evident from the figure...
The Lagrangian possesses the corresponding symmetries depending on which notion of torus you consider.
In the limit case of a torus with an infinite radius $R$, that is a cylinder, the two metrics coincide. This is the reason why you cannot see the problem just looking at the cylinder.
This is an interesting example where topology is not enough to fix physics. 
