Let $a:=\alpha$.

If $a\ne0$, then $X$ and $Y$ are normal by the Darmois--Skitovich theorem.

If $a=0$ and the distribution of $X$ is nondegenerate, then $U=X+Y$ and $V=X$ cannot be independent.

Let $a:=\alpha$.

If $a\ne0$, then $X$ and $Y$ are normal by the Darmois--Skitovich theorem.

If $a=0$ and the distribution of $X$ is nondegenerate, then $U=X+Y$ and $V=X$ cannot be independent.

If $a=0$ and the distribution of $X$ is degenerate, then $U=X+Y$ and $V=X$ are independent for any random variable $Y$.