I need to prove that given vector of commitments of length N contains N-1 commitments to zero (and one to an arbitrary number). 
More formally, given vector:
$$\textbf{a} = \begin{bmatrix}
    C(0, r_1) & C(0, r_2) & C(x, r_3) & ... & C(0, r_N)
 \end{bmatrix}$$
I want to prove that there is exactly one such commitment, that commits to *x*, rather than 0. Note that *x* is also private.  
I've seen the opposite to what I need to prove in [One-out-of-Many Proofs][1] paper, but still can't come up with what I need. 


  [1]: https://%20%20[1]:%20https://www.iacr.org/archive/eurocrypt2015/90560252/90560252.pdf