I would like to understand the nonlinear transform of Gaussian random variable that preserves Gaussianity better when there is no $x_3$ term such that there exists a nonlinear relationship between $x_1$ and $x_2$. I think $X \cdot Y$ cannot be then normally distributed.
I think it is a very rare case where the nonlinear relationship between two independent random variables maintains Gaussianity. I would like to show that where the substitution of zero by the mean $\eta$ from Robert's answer
Assume the conditional distribution of the two random variables given $z$ is always ${\mathbf N}(\eta,1)$ Suppose $X_1, \ldots, X_n$ are independent ${\mathbf N}(0,1)$ random variables, and ${\bf Y} = (Y_1, > \ldots, Y_n)$ is a vector-valued random variable independent of $X_1, > \ldots, X_n$ and supported on the sphere $Y_1^2 + \ldots + Y_n^2 = 1$. Then ${\bf X} \cdot {\bf Y} = X_1 Y_1 + \ldots + X_n Y_n \sim {\mathbf N}(\eta,1).$
where I am uncertain about the mean $\eta$ and how it carries there.
Practical case:
Take a plane where is a hole. Take a picture of the hole. Subtract bottom and top diameters from both sides, you get Gaussian distribution, which I think is because of the lightning effect on the unsteady surface (you never cannot have perfectly flat surface in practice). I call this process of irregularity the random variable with $\eta$.
Another challenge is to show that the factor $\eta$ comes in the play but here we do not focus on it because it comes there based on empirical measurements.
How can you maintain the Gaussianity between two independent random variables by a nonlinear transform?
This is nonsense as it stands. If $X_i$ has mean $0$ and $Y_i$ is independent of $X_i$, then $X_i Y_i$ has mean $0$. If you want to get a random variable of mean $\eta$, just add the constant $\eta$. That is:
Suppose $X_1, \ldots, X_n$ are independent ${\cal N}(0,1)$ random variables, and ${\bf Y} = (Y_1, \ldots, Y_n)$ is a vector-valued random variable independent of $X_1, \ldots, X_n$ and supported on the sphere $Y_1^2 + \ldots + Y_n^2 = 1$. Then $\eta + {\bf X} \cdot {\bf Y} = \eta + X_1 Y_1 + \ldots + X_n Y_n \sim {\cal N}(\eta,1)$.