  The existence of breathers (time-periodic and spatially localized lattice
vibrations) is well established for i) systems without acoustic phonon branches
and ii) systems with acoustic phonons, but also with additional symmetries
preventing the occurence of strains (dc terms) in the breather solution. The
case of coexistence of strains and acoustic phonon branches is solved (for
simple models) only for one-dimensional lattices.
  We calculate breather solutions for a two-dimensional lattice with one
acoustic phonon branch. We start from the easy-to-handle case of a system with
homogeneous (anharmonic) interaction potentials. We then easily continue the
zero-strain breather solution into the model sector with additional quadratic
and cubic potential terms with the help of a generalized Newton method. The
lattice size is $70\times 70$. The breather continues to exist, but is dressed
with a strain field. In contrast to the ac breather components, which decay
exponentially in space, the strain field (which has dipole symmetry) should
decay like $1/r^a, a=2$. On our rather small lattice we find an exponent
$a\approx 1.85$.
