Homogeneous Einstein metrics on the generalized flag manifold Sp(n)/(U(p)×U(n−p))Sp(n)/(U(p)×U(n−p))

We find the precise number of non-Kähler Sp(n)Sp(n)-invariant Einstein metrics on the generalized flag manifold M=Sp(n)/(U(p)×U(n−p))M=Sp(n)/(U(p)×U(n−p)) with n⩾3n⩾3 and 1⩽p⩽n−11⩽p⩽n−1. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.