(Throwing some ideas here, not sure how useful they are. Curve sketching is really all about noticing features and piecing every bits of clues together)
- The function is symmetric along x- and y-axes (clue: the function only has even powers of x's and y's), so looking at one quadrant will do. (I don't dare saying it's "even" for multivariate functions, as it doesn't quite make sense, but that's the idea anyway)
- If you rearrange terms, following presumably standard procedures for sketching contour lines (IDK what they are, really), you should get y is a quadratic in x^2, with some sort of vertical shifting controlled by what the function value takes. So I would expect some sort of quartic-looking curves.
- The x^2 term gives the "dimples" look to the quartic curves (this feels like a "experimental result from desmos").
- Stationary points are useful, obviously. Contour lines taken at different function values should "tend towards/away" stationary points, if that makes sense.