  Systematic effects caused by the Berry (geometric) phases in an
electric-dipole-moment experiment in an all-electric storage ring are
considered. We analyze the experimental setup when the spin is frozen and local
longitudinal and vertical electric fields alternate. Due to the Berry phases,
the spin rotates about the radial axis. The corresponding systematic error is
rather important while it can be canceled with clockwise and counterclockwise
beams. The Berry phases also lead to the spin rotation about the radial axis.
This effect can be canceled with clockwise and counterclockwise beams as well.
The sign of the azimuthal component of the angular velocity of the spin
precession depends on the starting point where the spin orientation is perfect.
The radial component of this quantity keeps its value and sign for each
starting point. When the longitudinal and vertical electric fields are joined
in the same sections without any alternation, the systematic error due to the
geometric phases does not appear. However, another systematic effect of the
spin rotation about the azimuthal axis takes place and has opposite signs for
clockwise and counterclockwise beams.
