  A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component
$\Omega$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of
the $\omega$-limit set of $F$ on $\Omega$ contains an isolated fixed point, (2)
there exists a univalent map $\Phi$ from $\Omega$ into $\mathbb C^2$
conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every
limit map of $\{F^{\circ n}\}$ on $\Omega$ has one-dimensional image. In this
paper we prove the existence of parabolic cylinders for an explicit class of
maps, and show that examples in this class can be constructed as compositions
of shears and overshears.
