Can $f(X)\times E+g(X)$ be a Gaussian if $E$ is a Gaussian for non-constant $f$ and $g$? It is well-known that $aE+b$ preserves the normality of $E$ when $a$ and $b$ are constants. However it is unclear whether this can happen when said constants become (possibly) dependent random variables.
So for simplicity, let $E\sim N(0,1)$ and $X$ is a continuous random variable independent of $E$, I want to know if there is some non-constant functions $f(x)$ and $g(x)$ such that $f(X)\times E+g(X)\sim N(0,1)$?

Generally the answer is no, loosely understanding "non-constant" also bans trivialities involving sets of probability zero.  But when $X$ also has a standard Normal distribution and $|\rho|\lt 1,$ notice that
$$Y = \rho\operatorname{sgn}(X) E + \sqrt{1-\rho^2}\,X$$
also has a standard Normal distribution.  The coefficient function $f(x) = \rho\operatorname{sgn}(x)$ and $g(x) = \sqrt{1-\rho^2}\,x$ are almost surely non-constant.
This claim can be demonstrated by noting (i) the transformation $(x,e)\to (x,\operatorname{sgn}(x)e)$ is almost surely differentiable and one-to-one; (ii) where it is differentiable, its Jacobian is $1;$ and (iii) the bivariate Normal density is unchanged by negating $e.$
BTW, an analogous construction applies to any bivariate Normal variable $(X,E)$ with zero mean.
