  We prove some general results about the relation between the 1-cocycles of an
arbitrary Lie algebroid $A$ over $M$ and the leaves of the Lie algebroid
foliation on $M$ associated with $A$. Using these results, we show that a
${\cal E}^1(M)$-Dirac structure $L$ induces on every leaf $F$ of its
characteristic foliation a ${\cal E}^1(F)$-Dirac structure $L_F$, which comes
from a precontact structure or from a locally conformal presymplectic structure
on $F$. In addition, we prove that a Dirac structure $\tilde{L}$ on $M\times
\R$ can be obtained from $L$ and we discuss the relation between the leaves of
the characteristic foliations of $L$ and $\tilde{L}$.
