Stability analysis for the pseudo-Riemannian geodesic flows of step-two nilpotent Lie groups

The paper proves that on a generic (maximal-dimensional) coadjoint orbit O the Lie–Poisson equation reduces to a linear constant-coefficient system dY/dt = j(Y_z)|_O Y, that O contains a unique equilibrium Y_v+Y_z with Y_v in ker j(Y_z), and that the Williamson type is determined by the eigenvalues of j(Y_z) itself; see the restriction (3.6) and the uniqueness statement immediately following it, together with Proposition 2.1 giving O = Y + Im j(Y_z) and T_Y O = Im j(Y_z) . The main result (Theorem 3.4) then reads off the elliptic/hyperbolic/focus–focus counts from the Cartan-subalgebra representatives (3.3)–(3.5) of s o(p,q) under the rank-2n assumption . The candidate solution reproduces exactly this block-by-block eigenvalue counting: 2×2 rotation blocks give elliptic pairs, 2×2 symmetric off-diagonal blocks give hyperbolic pairs, and the coupled 4×4 blocks yield focus–focus quadruples. The resulting formulas match Theorem 3.4 verbatim, and the use of (3.6) to identify the linearization is the same as in the paper. Hence both are correct and follow substantially the same proof strategy. For background on Williamson types, see §3.1 of the paper .