Complete description of invariant Einstein metrics on the generalized flag manifold $SO(2n)\mathrm{/}U(p)\times U(n-p)$

We find the precise number of non-K"ahler $SO(2n)$-invariant Einstein metrics on the generalized flag manifold $M=SO(2n)\mathrm{/}U(p)\times U(n-p)$ with $n\ge 4$ and $2\le p\le n-2$. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.