$Z=X+Y$ is Gaussian, $X$ & $Y$ are not, $X$ & $Y$ are independent. Example?

Question

Suppose $X$, $Y$ and $Z$ are random variables, with $Z=X+Y$.

Suppose further that $Z$ is Gaussian, $X$ and $Y$ are non-Gaussian, and that $X$ and $Y$ are independent.

What distribution could $X$ and $Y$ have? Any example would do!

Answer 1

No such $X$ and $Y$ exist.

A well-known characterization of the Gaussian distribution is given by Cramer's theorem:

Theorem. If the sum $X+Y$ of the random variables $X$ and $Y$ is normally distributed and these variables are independent, then each of $X$ and $Y$ are normally distributed.