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

We find the precise number of non-K"ahler $Sp(n)$-invariant Einstein metrics on the generalized flag manifold $M=Sp(n)\mathrm{/}(U(p)\times U(n-p))$ with $n\ge 3$ and $1\le p\le n-1$. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.