What is the symmetry group of a sum of n square doublets? $X= (a+b)^2+(c+d)^2+(e+f)^2...$

Question

Clear the symmetry group of $a^2+b^2+...=C$, where $C$ is a constant, are the circle and its higher-dimensional versions.

But what is the symmetry group given by

$$ (a+b)^2+(c+d)^2=C $$

and

$$ (a+b)^2+(c+d)^2+(e+f)^2=C $$

and finally, in the n case

$$ (a+b)^2+(c+d)^2+(e+f)^2+...=C $$