In fact, for any x in the interval 4/3 < x < 5 we can write , with the last matrix having rank 1.
In fact, I suspect that the only case where the answer is unique will be when C is a multiple of the identity. In that case, the unique solution is C = C + 0. For any other positive definite matrix C, we can write , where is the smallest eigenvalue of C. Then is positive definite, and is positive semi-definite. Of course, it may not have minimal rank. But even if it does, then (as in the numerical example above) I wouldn't expect that to be the unique solution.