The elliptic coordinate system for the plane, which transforms the coordinates $(\mu,\nu)$ to the cartesian $(x,y)$ via \begin{align} x&=a\cosh(\mu)\cos(\nu)\\ y&=a\sinh(\mu)\sin(\nu), \end{align} is a useful way to describe elliptical geometries in the plane (and, through its oblate and prolate extensions, in three-dimensional space). In this geometry, the surfaces of constant $\mu$ form a family of confocal ellipses of variable eccentricity, that start close to the interval between $(-a,0)$ and $(a,0)$, and approach circles as $\mu$ grows.
It was recently suggested to me, as a possible resolution to this question, that it might be possible to build an orthogonal coordinate system for the plane whose surfaces of constant coordinate $\mu$ are also ellipses of a constant eccentricity and variable foci, i.e. such that $$ \mu=\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}, $$ with a coordinate $\nu$ built so its lines of constant $\nu$ are orthogonal to the constant-$\mu$ ellipses.
Is this possible? If not, why not? If it is, are there analytical expressions for $\nu$? How does the laplacian look like in this coordinate system?