  We study a model of Josephson layered structure which is characterized by two
peculiarities: (i) superconducting layers are thin; (ii) due to suppression of
superconducting states in superconducting layers the current-phase relation is
non-sinusoidal and is described by two sine harmonics. The governing equation
is a nonlocal generalization of double sine-Gordon (NDSG) equation. We argue
that the dynamics of fluxons in the NDSG model is unusual. Specifically, we
show that there exists a set of particular velocities for non-radiating fluxon
propagation. In dynamics the presence of these ``priveleged'' velocitied
results in phenomenon of quantization of fluxon velocities: in our numerical
experiments a travelling kink-like excitation radiates energy and slows down to
one of these particular velocities, taking a shape of predicted 2-pi-kink. This
situation differs from both, double sine-Gordon local model and the nonlocal
sine-Gordon model, considered before. We conjecture that the set of these
velocities is infinite and present an asymptotic formula for them.
