  Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of
complex H\'enon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have
a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We
give a characterization of those H\'enon maps from the curve
$\mathcal{P}_{\lambda}$ that are small perturbations of a quadratic polynomial
$p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there
is an open disk of parameters in $\mathcal{P}_{\lambda}$ for which the
semi-parabolic H\'enon map has connected Julia set $J$ and is structurally
stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice local description:
inside a bidisk $\mathbb{D}_{r}\times \mathbb{D}_{r}$ it is a trivial fiber
bundle over $J_{p}$, the Julia set of the polynomial $p$, with fibers
biholomorphic to $\mathbb{D}_{r}$. The Julia set $J$ is homeomorphic to a
quotiented solenoid.
