  The Peierls distortions in a two-dimensional electron-lattice system
described by a Su-Schrieffer-Heeger type model extended to two-dimensions are
numerically studied for a square lattice. The electronic band is just
half-filled and the nesting vector is ($\pi/a$, $\pi/a$) with $a$ the lattice
constant. In contrast to the previous understanding on the Peierls transition
in two dimensions, the distortions which are determined so as to minimize the
total energy of the system involve not only the Fourier component with the
nesting wave vector but also many other components with wave vectors parallel
to the nesting vector. It is found that such unusual distortions contribute to
the formation of gap in the electronic energy spectrum by indirectly (in the
sense of second order perturbation) connecting two states having wave vectors
differing by the nesting vector from each other. Analyses for different system
sizes and for different electron-lattice coupling constants indicate that the
existence of such distortions is not a numerical artifact. It is shown that the
gap of the electronic energy spectrum is finite everywhere over the Fermi
surface.
