Function of two variables: Do symmetry tests exist?

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let's assume we are working with the function:

z=e-x2-y2

Plotting this function shows that there is a (global) maximum at (x, y) = (0, 0).

The plot clearly shows some sort of symmetry to the coordinate origin. Is there a correct term to describe this symmetry and more importantly, is there a way to check (analytically) whether a function of two variables has any symmetries?

Thankful for any input.