is attained at the nondegenerate point of maximum $y_c(x)$ ([@Sj82]).

\[ir1\] We have $\kappa (\Lambda _G)=\Lambda _{\Phi _G}$, where [$$\label{ir.2}
\Phi _G(x)=\mathrm{v.c.}_{\widetilde{y},\eta }-\Im \phi
(x,\widetilde{y})-\eta \cdot \Im \widetilde{y}+G(\Re
\widetilde{y},\eta ),$$]{} and the critical value is attained at a nondegenerate critical point. Here $\mathrm{v.c.}_{\widetilde{y},\eta }(...)$ means “critical value with respect to $\widetilde{y},\eta $ of ...”.

At a critical point we have $$\begin{aligned}
&\Im \widetilde{y}=\nabla _\eta G(\Re \widetilde{y},\eta ),&\\
&\frac{\partial }{\partial \Im \widetilde{y}}\Im \phi
(x,\widetilde{y})+\eta =0,&\\
&-\frac{\partial }{\partial \Re \widetilde{y}}\Im \phi
(x,\widetilde{y})+(\nabla _y G)(\Re \widetilde{y},\eta )=0.&\end{aligned}$$ If $f(z)$ is a holomorphic function, then [$$\label{ir.3}\frac{\partial }{\partial \Im z}\Im f=\Re 
\frac{\partial f}{\partial z}
,\ \frac{\partial }{\partial \Re z}\Im f=
\Im \frac{\partial f}{\partial z},$$]{} so the equations for our critical point become $$\begin{aligned}
\Im \widetilde{y}&=&\nabla _\eta G(\Re \widetilde{y},\eta ),\\
\eta &=&-\Re \frac{\partial \phi }{\partial \widetilde{y}}(x,\widetilde{y}),\\
-\nabla _yG(\Re \widetilde{y},\eta )&=&-\Im \frac{\partial \phi
}{\partial \widetilde{y}},\end{aligned}$$ or equivalently, $$(\widetilde{y},-\frac{\partial \phi }{\partial
  \widetilde{y}}(x,\widetilde{y}))=(\Re \widetilde{y},\eta )+iH_G(\Re
\widetilde{y},\eta ),$$ which says that the critical point $(\widetilde{y},\eta )$ is determined by the condition that $\kappa _T$ maps the point $(\widetilde{y},\widetilde{\eta })\in \Lambda _G$ to a point $(x,\xi
)$, situated over $x$. Clearly the critical point is nondegenerate. We check it when $G=0$: The Hessian matrix with respect to the variables $\Re y,\Im 