  Our main results are: (1) The complex a Lagrangian points of a non-complex
Lagrangian $2n$-dimensional submanifold $F:M\ra N$, immersed with parallel mean
curvature and with equal Kaehler angles into a Kaehler-Einstein manifold
$(N,J,g)$ of complex dimension $2n$, are zeros of finite order of
$\sin^2\theta$ and $\cos^2\theta$ respectively, where $\theta$ is the common
$J$-Kaelher angle. (2) If $M$ is a Cayley submanifold of a Calabi-Yau (CY)
manifold $N$ of complex dimension 4, then $\bigwedge^2_+NM$ is naturally
isomorphic to $\bigwedge^2_+TM$. (3) If $N$ is Ricci-flat (not necessarily CY)
and $M$ is a Cayley submanifold, then $ p_1(\bigwedge^2_+NM)=
p_1(\bigwedge^2_+TM)$ still holds, but $p_1(\bigwedge^2_-NM)-
p_1(\bigwedge^2_-TM)$ may describe a residue on the $J$-complex points, in the
sense of Harvey and Lawson. We describe this residue by a PDE on a natural
morphism $\Phi:TM \to NM$, $\Phi(X)=(JX)^{\bot}$, with singularities at the
complex points. We give an explicit formula of this residue in a particular
case. When $(N,I,J,K,g)$ is an hyper-Kaehler manifold and $M$ is an $I$-complex
closed 4-submanifold, the first Weyl curvature invariant of $M$ may be
described as a residue on the $J$-Kaehler angle at the $J$_Lagrangian points by
a Lelong-Poincar\'{e} type formula. We study the almost complex structure $\Jw$
on $M$ induced by $F$.
