  We prove the existence of time-periodic, small amplitude solutions of
autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of
Benjamin-Ono type in Sobolev class. The result holds for frequencies in a
Cantor set that has asymptotically full measure as the amplitude goes to zero.
At the first order of amplitude, the solutions are the superposition of an
arbitrarily large number of waves that travel with different velocities
(multimodal solutions). The equation can be considered as a Hamiltonian,
reversible system plus a non-Hamiltonian (but still reversible) perturbation
that contains derivatives of the highest order. The main difficulties of the
problem are: an infinite-dimensional bifurcation equation, and small divisors
in the linearized operator, where also the highest order derivatives have
nonconstant coefficients. The main technical step of the proof is the reduction
of the linearized operator to constant coefficients up to a regularizing rest,
by means of changes of variables and conjugation with simple linear
pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov
and Toland for standing water waves (ARMA 2005). Other ingredients are a
suitable Nash-Moser iteration in Sobolev spaces, and Lyapunov-Schmidt
decomposition.
  (Version 2: small change in Section 2).
