  We investigate the exact wavefunction as a single product of electronic and
nuclear wavefunction for a model conical intersection system. Exact factorized
spiky potentials and nodeless nuclear wavefunctions are found. The exact
factorized potential preserves the symmetry breaking effect when the coupling
mode is present. Additionally the nodeless wavefunctions are found to be
closely related to the adiabatic nuclear eigenfunctions. This phenomenon holds
even for the regime where the non-adiabatic coupling is relevant, and sheds
light on the relation between the exact wavefunction factorization and the
adiabatic approximation.
