The MAPLE and SINGULAR codes em1.mw, em1.in solve the Ernst-Maxwell equations up to linear terms with additional assumptions: {\cal E}(z_0)=\varphi(z_0)=0 and Df(z_0) ≠0 (see the beginning of Section 3 in the article). Another set of assumptions, {\cal E}(z_0)=\varphi(z_0)=Df(z_0)=0, DDf(z_0)≠0 and the ansatz (5.1) (see the article) are analysed in the MAPLE file em2.mw. The examples of balanced leading-order solutions with radial $\mcE$ (solutions to Eq. 5.10 for k=2, k=3) are presented in the MAPLE file em3.mw. With help of two remaining SINGULAR codes em4a.in, em4b.in we show that there do not exist a particular subclass of \varphi-dominated solutions (see the beginning of Subsection 5.3).