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We find the precise number of non-K\"ahler $Sp(n)$-invariant Einstein metricson the generalized flag manifold $M=Sp(n)/(U(p)\times U(n-p))$ with $n\geq 3$and $1\leq p\leq n-1$. We use an analysis on parametric systems of polynomialequations and we give some insight towards the study of such systems.

Homogeneous Einstein metrics on the generalized flag manifold Sp(n)/(U(p) x U(n-p))