There is some freedom to add some additive constants to the definition of $\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\alpha}_{3}$, which does not affect the validity of the linearised equations. Our choice ensures that $\tilde{\alpha}_{i}-\alpha_{i}$ vanishes at the origin, which is need later for gluing in the Taub-NUT type metric on $\mathbb{C}^{3}$. A more quantitative statement is: