I was reading the following paper

about the Taub-NUT metric, and I don't understand the origin of equation (2.55). Here is the (more or less self-contained) description of the issue:

the authors have constructed an orthonormal tetrad $(l,n,m,\overline{m})$, in the coordinate system $(x^{1},r,x^{3},x^{4})$. At some point the vector field $m$ has the form

$m=\overline{\rho}\hskip 1pt(Y_{1}\: , \: 0 \: , \: Y_{2} \: , \: iY_{2})$

where $\rho$ is a function of $r$, and $Y_{2}=Y_{2}(x^{3},x^{4})$ is a real function. The coordinate freedom still available is the coordinate transformation of the form

$\zeta'=g(\zeta)$

where $\zeta:=x^{3}+ix^{4}$.

QUESTION: the authors say, "it is possible to choose such a coordinate transformation $g$ so that $Y_{2}=\frac{1}{\sqrt{2}}+\frac{M}{2\sqrt{2}}((x^{3})^{2}+(x^{4})^{2})$", where $M$ is some constant. How do they know that? How could I see that? In general, what kind of calculation do I have to do for $Y_{2}$ to see what kind of $g$ I could pick?

Thank you for any hints