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

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