A Liouville theorem argument shows that the solution $v^{ij}$ and $w$ to the linear system of equation (2.2)(2.3)(2.4) is unique, in the sense that if another solution differs from it by functions with some power law decay at infinity, then the two solutions agree. The key point is to analyse the difference of the two solutions, and observe that now there is no forcing term in the distributional equation, so the $\Delta_{a}$-harmonicity extend across the discriminant locus.