It will turn out in the next few Sections that $v$ and $w^{p\bar{q}}$ have logarithmic growth at infinity bounded away from $S$. If we restrict to solutions to (4.4)(4.5)(4.6) with the same growth properties, then $v$ and $w^{p\bar{q}}$ are unique up to additive constants. The choices of these constants are not completely canonical, related to the philosophy that the Ooguri-Vafa type metrics are only effective descriptions admitting a certain amount of small fluctuation.