  This work deals with a scalar nonlinear neutral delay differential equation
issued from the study of wave propagation. A critical value of the coefficients
is considered, where only few results are known. The difficulty follows from
the fact that the spectrum of the linear operator is asymptotically closed to
the imaginary axis. An analysis based on the energy method provides new results
about the asymptotic stability of the constant and periodic solutions. A
complete analysis of the stability diagram is given. Lastly, existence of
periodic solutions is discussed, involving a Diophantine condition on the
period.
