Energy of singular logarithmic potential

We saw in a physics/astronomy course that we can model a path in a 2D-galaxy with the following singular logarithmic potential

$$ Phi_{q}(x, y)=frac{v_{0}^{2}}{2} ln left[x^{2}+left(frac{y}{q}right)^{2}right], $$

where $(x, y)$ is the coördinate in the 2D-space and $v_0$ is the initial velocity of the particle (e.g. star) whose orbit we want to model. If $q=1$, then we have a axial symmetrical potential and when $q<1$, then we have a bar-shaped galaxy. Now they say we can set $v_0=1$ if we choose the right units for time and distance. I think I get that, it is sort of like the natural units, where $c=1$, but now we take $v_0=1$.

But they also say that this potential in particular gives the same orbitshape for different energies. How can you know this physically or can you calculate this mathematically from this potential?

Thanks in advance!