give examples of surfaces containing smooth curves of self-intersection number $d^2$ which are not embeddable in the plane, although they are fibered by submersions whose restrictions to the curves are meromorphic functions of a degree multiple of $d$. Once this is done, we give examples which satisfy the extra condition of our Theorem but have only one or two fibrations and do not embed them in the plane.

Separating branches and examples with 3 fibrations
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We will use the following construction. Let us consider a curve H with an ordinary singularity $P$ with $m$ branches $L_1,\dots,L_m$. For each branch $L_j$ we take a neighborhood $V_j$ which is biholomorphic to a bidisc $D_j$ by means of a biholomorphism $\phi_j:D_j\rightarrow V_j$ ; we assume that $\delta L_j \cap V_i =\emptyset$ for all $i\ne j$. We fix a neighborhood $V$ of $H\setminus \cup_1^m L_j$. Finally we take the disjoint union of $V$ with all th $D_j$, and glue $D_j$ to $V$ using the restriction of the map $\phi_j$ to $\phi_j^{-1}(V\cap V_j)$. In this way the union of the sets $V_j$, which contains $P$, is replaced by $m$ copies of the bidisc, and there is a new curve $H^{\prime}$ replacing $H$ inside a 