 Solution

The $O(2)$ rotation symmetry requires the quadratic dependence on $({\varphi }_2,{\varphi }_3)$ to be proportional to ${\varphi }_2^2+{\varphi }_3^2$, soThe quartic terms involving only that pair must be proportional to $({\varphi }_2^2+{\varphi }_3^2{)}^2$, and the terms coupling it to ${\varphi }_1$ must be proportional to ${\varphi }_1^2({\varphi }_2^2+{\varphi }_3^2)$. With the convention that the symmetric double sum counts off-diagonal terms twice, this giveswhile ${\mu }_1^2$, $g_{11}$, and the common mixed coupling are unrestricted. The stated ${\mathbb{Z}}_2$ symmetry imposes no further relation because every term is already even in ${\varphi }_1$.