  We realize specific classical symmetric spaces, like the semi-K\"ahler
symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag
manifolds. These realizations enable us to describe these cotangent bundles'
geodesics and Lagrangian submanifolds.
  As a final application, we present the first examples of vector bundles over
simply connected manifolds with nonnegative curvature that cannot accommodate
metrics with nonnegative sectional curvature, even though their associated unit
sphere bundles can indeed accommodate such metrics. Our examples are derived
from explicit bundle constructions over symmetric flag spaces.
