In this paper [N. Ghoussoub](https://www.degruyter.com/document/doi/10.1515/crll.1991.417.27/html),
the author claims the following version of Marino–Prodi perturbation, that is :


Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K\_c$ (that is, the set of critical points with critical value $c$) on which $d^2f$ is a Fredholm operator, for any $\epsilon\_1>0$ and $\epsilon\_2>0$ small enough, there exist a function $g\in C^2(H, \mathbb{R})$ such that :


* $\|f-g\|\_{C^2}<\epsilon\_1.$
* $f(x)=g(x)$ if $x$ outside $K^{2\epsilon\_2}$, that is, $2\epsilon\_2$-neighborhood of $K$.
* All the critical points of $g$ in $K^{\epsilon\_2}$ are non-degenerate and finite in number.
* Moreover, If $f$ is $G$-invariant functional for a a topological group $G$ on $H$, then $g$ can be chosen to be $G$-invariant.


In particular, if $G=Z\_2$, $f$ is even, hence $g$ is even.


The proof in [Solimini](https://link.springer.com/content/pdf/10.1007/BF01171757.pdf) for the first three conclusions is given here:


* First construct a smooth cut-off function $\phi(x)$, $\phi(x)$ equals $1$ on $K^{\epsilon\_2}$,
equals $0$ outside $K^{2\epsilon\_2}$.
* Set $g(x)=f(x)+\phi(x)y.x$, then by Sard-Smale Theorem, choosing a small $y$ such that $-y$ is a regular value of $df$, hence we complete the proof of the first three conclusions.


However, my question is how to construct a $G$-invariant $g$ when $f$ is $G$-invariant? In particularly, when $G=Z\_2$.



A question about Marino–Prodi perturbation
The proof for $G=Z\_2$ is given in [jde2016](http://dx.doi.org/10.1016/j.jde.2016.09.018) Appendix B Proposition B3.


