  We investigate families of soliton solutions in a spin-orbit coupled
Bose-Einstein condensate embedded in an optical lattice, which bifurcate from
the nearly flat lowest band. Unlike the conventional gap solitons the obtained
solutions have the shape well approximated by a Wannier function (or a few
Wannier functions) of the underlying linear Hamiltonian with amplitudes varying
along the family and with nearly constant widths. The Wannier solitons (WSs)
sharing all symmetries of the system Hamiltonian are found to be stable. Such
solutions allow for the construction of Wannier breathers, that can be viewed
as nonlinearly coupled one-hump solitons. The breathers are well described by a
few-mode model and manifest stable behavior either in an oscillatory regime
with balanced average populations or in a self-trapping regime characterized by
unbalanced atomic populations of the local potential minima (similarly to the
conventional boson Josephson junction), with the frequencies controlled by the
inter-atomic interactions.
