  We study the orbit behavior of a four dimensional smooth symplectic
diffeomorphism $f$ near a homoclinic orbit $\Gamma$ to an 1-elliptic fixed
point under some natural genericity assumptions. 1-elliptic fixed point has two
real eigenvalues out of unit circle and two others on the unit circle. Thus
there is a smooth 2-dimensional center manifold $W^c$ where the restriction of
the diffeomorphism has the elliptic fixed point supposed to be generic (no
strong resonances and first Birkhoff coefficient is nonzero). Moser's theorem
guarantees the existence of a positive measure set of KAM invariant curves.
$W^c$ itself is a normally hyperbolic manifold in the whole phase space and due
to Fenichel results every point on $W^c$ has 1-dimensional stable and unstable
smooth invariant curves forming two smooth foliations. In particular, each KAM
invariant curve has stable and unstable smooth 2-dimensional invariant
manifolds being Lagrangian. The related stable and unstable manifolds of $W^c$
are 3-dimensional smooth manifolds which are supposed to be transverse along
homoclinic orbit $\Gamma$. One of our theorems presents conditions under which
each KAM invariant curve on $W^c$ in a sufficiently small neighborhood of
$\Gamma$ has four transverse homoclinic orbits. Another result ensures that
under some Birkhoff genericity assumption for the restriction of $f$ on $W^c$
saddle periodic orbits in resonance zones also have homoclinic orbits though
its transversality or tangency cannot be verified directly.
