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We prove some general results about the relation between the 1-cocycles of anarbitrary Lie algebroid $A$ over $M$ and the leaves of the Lie algebroidfoliation 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 itscharacteristic foliation a ${\cal E}^1(F)$-Dirac structure $L_F$, which comesfrom a precontact structure or from a locally conformal presymplectic structureon $F$. 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 ofthe characteristic foliations of $L$ and $\tilde{L}$.Comment: 25 page

Lie algebroid foliations and $\Script E$1(M)-Dirac structures