I need to convert a mathematica expression from describing the geometry of a 2D ellipse in cartesian coordinates to an ellipse cutting normal to the rotating minor axis.

So far, it appears that Wolfram Mathematica plot function is restrained to plots in 3D Cartesian or polar systems,

I have tried rthe following

TransformedField["Cartesian" -> {{"OblateSpheroidal", a}}, 
  x^2/4 + y^2/3, {x, y, z} -> {\[Xi], \[Eta], \[Phi]}] // Simplify

This returns -\frac{1}{24} a^2 \sin ^2(\eta ) \cosh ^2(\xi ) (\cos (2 \phi )-7)

But I need the parametric form hat will allow me to plot using ParametricPlot3D.

trans = CoordinateTransformData[{{"OblateSpheroidal", 1}, 3} -> 
"Cartesian", "Mapping"]
ParametricPlot3D[{trans[{1, \[Eta], \[CurlyPhi]}]}, {\[Eta], 0, 
Pi}, {\[CurlyPhi], -\[Pi], \[Pi]}, PlotStyle -> Opacity[.5], 
ImageSize -> Medium]