This is a tentatve (since I am not sure I am parsing your query correctly) comment, but too long for that format. I take it that you are asking about the existence of diffeomorphisms of the interior of an ellipse which are area-preserving and preserve the foliation consisting of dilations of the boundary. The answer to this question is--yes, there are a multitude. The rather imprecise formulation is that if you specify two distinct radial vectors on the one hand and two suitable curves from the centre to the boundary, then there is a unique such diffeomorphism which maps the former pair of curves to the latter.
There are two practical problems which are (from a mathematical point if view) identical to yours (stripped of the specifics)
thermodynamics--identifying possible sysems of isotherms and adiabats of a thermodynamical system;
cartography--identifying possible systems of parallels and meridians of area-equivalent map projection.
In both cases, the rough and ready answer is as follows: you can choose one of the systems at random (think hyperbolae $xy=c$ for isotherms ($T=pV$!) and $y=c$ for parallels). One can then choose two adiabats, resp., two meridians at will and this determines the whole configuration.
There doesn't seem to be much literature on this but for the thermodynamics you could try the arXiv papers 1102.1540, 1108.4758.
Edit: If we use new variables $u$ and $v$ where $u=(b^2 x^2+y^2/b^2)/2$ and $v$ is the angle $CAB$ where $A$ is the origin, $B$ is $(1,0)$ and $C$ is $(bx,y/b)$ (these are just polar coordinates in the circle case $b=1$ adjusted to make the transformation area-preserving ), then the required transformations have the form $$(u,v) \mapsto (u,v+f(u))$$ for suitable functions $f$. (This is basically the solution of Robert Bryant but without using heavy machinery).