  In recent work, Baird et al. have generalized the definition of the Maslov
index to paths of Grassmannian subspaces that are not necessarily contained in
the Lagrangian Grassmannian [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R.
Marangell, {\it Generalized Maslov indices for non-Hamiltonian systems}, SIAM
J. Math. Anal. {\bf 54} (2022) 1623-1668]. Such an extension opens up the
possibility of applications to non-Hamiltonian systems of ODE, and Baird and
his collaborators have taken advantage of this observation to establish
oscillation-type results for obtaining lower bounds on eigenvalue counts in
this generalized setting. In the current analysis, the author shows that
renormalized oscillation theory, appropriately defined in this generalized
setting, can be applied in a natural way, and that it has the advantage, as in
the traditional setting of linear Hamiltonian systems, of ensuring monotonicity
of crossing points as the independent variable increases for a wide range of
system/boundary-condition combinations. This seems to mark the first effort to
extend the renormalized oscillation approach to the non-Hamiltonian setting.
