  We present what we believe to be the first known example of an exact
quasiperiodic localized stable solution with spatially symmetric
large-amplitude oscillations in a non-integrable Hamiltonian lattice model. The
model is a one-dimensional discrete nonlinear Schr\"odinger equation with
alternating on-site energies, modelling e.g. an array of optical waveguides
with alternating widths. The solution bifurcates from a stationary discrete gap
soliton, and in a regime of large oscillations its intensity oscillates
periodically between having one peak at the central site, and two symmetric
peaks at the neighboring sites with a dip in the middle. Such solutions, termed
'pulsons', are found to exist in continuous families ranging arbitrarily close
both to the anticontinuous and continuous limits. Furthermore, it is shown that
they may be linearly stable also in a regime of large oscillations.
