Is it possible to construct two random variables $X, Y$ both of them assuming exactly two non-zero values which are uncorrelated, i. e. $\mathbf{E}[X \, Y] = \mathbf{E}[X]\,\mathbf{E}[Y]$, but not independent?

If that is not possible, what is the simplest example of non-zero discrete random variables which are uncorrelated but not independent?

Thanks a lot!