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Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.

0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$. (Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)

1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$. For any $p\in M$, there exists a lagrangian submanifold of $(M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.

Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,
the restriction of $T_x \mathcal{F}x=\textnormal{span}\left(\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x\right)$, for any \mathcal{F}$on$x\in U$,is generated by$\frac{\partial}{\partial\phi_{n+1}},\ldots,\frac{\partial}{\partial\phi_{2n}}$, and consequently$\phi
{n+1}=\ldots=\phi_{2n}=0$\phi_{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.

This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.

2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.

This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.

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Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.

0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$. (Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)

1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$. For any $p\in M$, there exists a lagrangian submanifold of $(M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.

Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,
$T_x \mathcal{F}x=\frac{\partial}{\partial\phix=\textnormal{span}\left(\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x$x\right)$, for any$x\in U$, and consequently$\phi
{n+1}=\ldots=\phi_{2n}=0$is a local lagrangian submanifold of$(M,\omega)$passing through$p$and transverval to$\mathcal{F}$. This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system$d\phi_1,\ldots,d\phi_n$of$1$-forms which locally generates the distribution corresponding to the lagrangian foliation$\mathcal{F}$. 2. A relative globalization. If$L$, a lagrangian submanifold of$(M,\omega)$, is transversal to$\mathcal{F}$, then there exists a diffeomorphism$f$from an open neigborhood of$L$in$M$onto an open set in$T^*L$such that:$f|_L$is the zero section of$\tau_L^{\ast}:T^{\ast}L\to L$,$f_{\ast}\omega$is the canonical symplectic on$T^{\ast}L$, and$f$takes the leaves of$\mathcal{F}$in the fibers of$\tau^{\ast}_L$. This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein. 4 deleted 13 characters in body; deleted 12 characters in body Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful. 0.Setting and specification of the terminology. In a symplectic$2n$-dimensional manifold$(M,\omega)$, let be given a lagrangian foliation$\mathcal{F}$, i.e. a foliation of$M$whose leaves are lagrangian w.r.t.$\omega$. (Instead, I mean a lagrangian fibration of$(M,\omega)$as a surjective summersion$f:M\to B$whose fibers are lagrangian w.r.t.$\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).) 1.Local Existence of lagrangian submanifolds transversal to$\mathcal{F}$. For any$p\in M$, there exists a lagrangian submanifold of$(M,\omega)$which passes through$p$and is transversal to$\mathcal{F}$. Infact, for any$p\in M$, there exists a chart$(U,\phi)$for$M$centered at$p$, such that:$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,$T_x\mathcal{F}x=\textrm{span}\left(\frac{\partial}{\partial\phiT_x \mathcal{F}x=\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x\right)$x$, for any $x\in U$,
and consequently $\phi {n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.

This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.

2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.

This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.

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