Is it possible to find (necessary and sufficient) conditions on a general $n\times n$ Hermitian matrix $A$, such that its extremal eigenvectors (the eigenvectors corresponding to the maximum and minimum eigenvalues of $A$) have (pairwise) equal magnitude entries?

As an example, consider the case $n=2$. A basis in the space of $2\times 2$ Hermitian matrices is given by the identity matrix $\mathbb I_2$ together with the three Pauli matrices \begin{equation} \sigma_1 = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\;,\qquad \sigma_2 = \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)\;,\qquad \sigma_3 = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\;.\qquad \end{equation} The eigenvectors of the general $2\times 2$ Hermitian matrix $A=\alpha \mathbb I_2 + \beta \sigma_1 + \gamma \sigma_2 + \delta \sigma_3$ ($\alpha,\beta,\gamma,\delta\in\mathbb R$) are \begin{equation} \left(-\frac{\sqrt{\beta^2+\gamma^2+\delta^2}-\delta}{\beta+i\gamma},\,1\right)^T\;,\qquad \left(\frac{\sqrt{\beta^2+\gamma^2+\delta^2}+\delta}{\beta+i\gamma},\,1\right)^T\;. \end{equation} Therefore, if $\delta=0$, then the extremal eigenvalues have the desired property.

As an other example, consider the case $n=3$. A basis in the space of $3\times 3$ Hermitian matrices is given by the identity matrix $\mathbb I_3$ together with the eight Gell-Mann matrices $\{\lambda_i\}_{i=1,\dots,8}$. For instance, consider the subspace spanned by $\lambda_4,\lambda_6,\lambda_7$, i.e. set $A=\alpha \lambda_4 + \beta \lambda_6 + \gamma \lambda_7$. The extremal eigenvectors of $A$ are \begin{equation} \left(-\frac{\alpha }{\sqrt{\alpha ^2+\beta ^2+\gamma ^2}},-\frac{\beta -i \gamma }{\sqrt{\alpha ^2+\beta ^2+\gamma ^2}},\,1\right)^T\;,\qquad \left(\frac{\alpha }{\sqrt{\alpha ^2+\beta ^2+\gamma ^2}},\frac{\beta -i \gamma }{\sqrt{\alpha ^2+\beta ^2+\gamma ^2}},\,1\right)^T\;. \end{equation} The desired property holds in this case. However, I don't know what is the maximal subspace on which it holds.

In conclusion, the question asks for general conditions on a $n\times n$ Hermitian matrix $A$, for $n>2$, such that the extremal eigenvectors of $A$ have equal magnitude entries.