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We find the precise number of non-Kähler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ 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.

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p) × U(n − p)