  The formation and local symmetry of spin-lattice polaron has been
investigated semiclassically in the planar Holstein t-J-like models within the
exact diagonalization method. Due to the interplay of strong correlations and
electron-lattice interaction, the doped hole may either move freely or lead to
the localized spin-lattice distortion and form a Holstein polaron. The
formation of polaron breaks the translational symmetry by suppression of
antiferromagnetic correlations and inducement of ferromagnetic correlations
locally. Moreover, the breaking of local rotational symmetry around the polaron
has been shown. The ground state is generically a parity singlet and the first
excited state maybe a parity doublet. Further consequences of the density of
states spectra for comparison with future STM experiments are discussed.
