  The predominant source of nonlinearity in most existing accelerators are
sextupoles, which introduce nonlinear resonances. When the horizontal tune of
an accelerator is near such a resonance line ($n\nu_x$), stable fixed points
(SFPs) may appear in the horizontal phase space to form a second closed orbit
different from the "zero" closed orbit. The stable islands in phase space
surrounding the SFPs are also referred as transverse resonance island "buckets"
(TRIBs). However, the TRIBs are not always present near a resonance line in a
storage ring. Necessary conditions for TRIBs formation will be discussed in
this paper. A 6-GeV lattice with the horizontal tune near a 3rd-order resonance
line at $3\nu_{x}$=50 is then designed for the Cornell Electron Storage Ring
(CESR). After loading this lattice into CESR, the TRIBs are observed while
adjusting the horizontal tune near the 3rd-order resonance, consistent with
particle tracking simulations. TRIBs rotation in the horizontal phase space are
also experimentally demonstrated with adjustment of a sextupole knob. These
results are in good agreement with the theoretical calculation.
