Finding the unitary matrices U and V of an SVD product using the product UV of the unitary matrices?

Question

Given the SVD decomposition of a know matrix $M$ such as : $M = U S V^t $

Now consider the matrices $U$, and $V$ are unknown, and only their matrix product $U V^t$ and the matrices $M$ and $S$ are known, how we can find the matrices $U$ and $V$ ? I thought using polar decomposition, hints are welcome.

P.S: $M$ is a real valued square matrix

Thanks.

Answer 1

Let $M=I$, $S=I$, and $V=U$. Then $I=UIU^t$ holds, but knowing $UU^t=I$, we still no knowing about $U$ (except that it's a unitary).