I want to know if there exist a function $g$ and $\mu \ne \mu’$ such that $$ \int g(x)\phi(x;\mu)\,dx = \int g(x)\phi(x;\mu’)\, dx$$ Where $\phi(x;\mu)$ is the pdf of normal random variable with mean $\mu$ and unit variance.
All I can find is that if $g$ is symmetric — $g(x) = g(-x)$, then the above condition holds.
Could there be other functions that satisfy the above equation?
There is such a function $g$. Take $g(x)=x-\frac 1{\mu+\mu'} x^{2}$. The case $\mu+\mu'=0$ is left to you.