  We propose an analytically solvable sextic potential model with non-trivial
soliton solutions connecting the trivial vacua. The model does not respect
parity symmetry, and like $\phi^4$ theory has two minima. The soliton solutions
and the consequent results are obtained in terms of the Lambert W function,
i.e., the inverse function of $f(W) = We^W$. They have power-law asymptotics at
one spatial infinity and exponential asymptotics at the other. We compare the
solution with the kink of $\phi^4$ theory, which preserves the parity symmetry
and has exponential asymptotics at both spatial infinities. Moreover, we study
the full spectrum (bound and continuum states) of boson and fermion fields in
the presence of the proposed soliton. We consider two types of coupling for the
boson-soliton interaction and Yukawa coupling for the fermion-soliton
interaction. Most results are derived analytically. This property renders the
model a fertile ground for further study, including parity breaking related
phenomena and long-range soliton-soliton interactions.
