Given $$A_{i,j,k} X_j^* X_k + C_i = 0$$ where $𝐴_{i,j,k}$ and $𝐶_i$ are arbitrary complex numbers for all $𝑗, 𝑘$ which are $𝑁$-dimensional indices and $i$ which is an $m$-dimensional index where $m<N$. Note $*$ is complex conjugation and there is a summation for repeated indices. Are there conditions on $A$ and $C$ to guarantee the existence of a solution to this set of equations?