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LINKAGE
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LEMMA The distinct real eigenvalues of $\bar{A}$ which differ from the
$\bar{n}$ values $\bar{\gamma}_i$
are the solutions of
\[
 g_{\bar{\mu}}(\lambda) = t.
\]
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denote the set of Lagrange multipliers of $\bar{\mu}(\cdot)$ with
respect to $S_{\bar{n}}$ by $\bar{\Lambda}$ (easy case),  
\[\bar{\lambda}_1 > \bar{\lambda}_2 > \ldots > \bar{\lambda}_m  .\]
the set of critical points of $g_{\bar{\mu}}(\cdot)$
is
$\bar{\Lambda}$ 
\[
\bar{\mu}_1=g_{\bar{\mu}}(\bar{\lambda}_1) > \bar{\mu}_2 =
g_{\bar{\mu}}(\bar{\lambda}_2)
>\ldots>\bar{\mu}_m=g_{\bar{\mu}}(\bar{\lambda}_m),
\]
stationary values, points of $\bar{\mu}(\cdot)$ on $S_{\bar{n}}$ 
\[ \bar{\mu}_i = \bar{\mu}(\bar{x}^{\bar{\lambda}_i}),~
i=1,2,\ldots,m\]
\[\bar{x}^{\bar{\lambda}_i}=(\bar{D}-\bar{\lambda}_i\bar{I})^{-1}\bar{\eta},~
i=1,2,\ldots,m,\]
$\bar{\eta}$ is $\bar{n}$-vector with $i^{th}$ component
$\sqrt{\bar{\alpha}_i\bar{\beta}_i}$.