  Kuramoto Networks contain non-hyperbolic equilibria whose stability is
sometimes difficult to determine. We consider the extreme case in which all
Jacobian eigenvalues are zero. In this case linearizing the system at the
equilibrium leads to a Jacobian matrix which is zero in every entry. We call
these equilibria completely degenerate. We prove that they exist for certain
intrinsic frequencies if and only if the underlying graph is bipartite, and
that they do not exist for generic intrinsic frequencies. In the case of zero
intrinsic frequencies, we prove that they exist if and only if the graph has an
Euler circuit such that the number of steps between any two visits at the same
vertex is a multiple of 4. The simplest example is the cycle graph with 4
vertices. We prove that graphs with this property exist for every number of
vertices N larger than 5 and that they become asymptotically rare for N large.
Regarding stability, we prove that for any choice of intrinsic frequencies, any
coupling strength and any graph with at least one edge, completely degenerate
equilibria are not Lyapunov stable. As a corollary, we obtain that stable
equilibria in Kuramoto Networks must have at least one strictly negative
eigenvalue.
