The equipotential curves of two parallel equal-density line charges (in a plane perpendicular to the line charges) are known to be Cassini ovals. This is because the potential at a point a distance $r_1$ from one line charge and $r_2$ from the other line charge is
$$V = \frac{\lambda}{2\pi\epsilon_0}\left(\log\left(\frac{r_1}{r_0}\right)+\log\left(\frac{r_2}{r_0}\right)\right)=\frac{\lambda}{2\pi\epsilon_0}\log\left(\frac{r_1r_2}{r_0^2}\right)$$
which means that the equipotential curves require that $r_1r_2$ is constant, the same condition that generates Cassini ovals.
However, there doesn't seem to be any analogous well-known geometric form for the equipotentials generated by two equal point charges. The equipotential curves in a plane coplanar with the charges require that $\frac{1}{r_1}+\frac{1}{r_2}$ be constant. Here are the only things I can find out about these curves:
The implicit equipotential curves are of degree eight in Cartesian coordinates (from "Families of Ovals and their Orthogonal Trajectories," D.F. Lawden, The Mathematical Gazette, link: https://www.jstor.org/stable/3620950)
They look similar to Cassini ovals, but are not (see https://en.wikipedia.org/wiki/Implicit_curve#Applications_of_implicit_curves)
Can we classify these equipotential curves as instances of a single geometric form in a similar fashion to the line-charge case?
