I am trying to understand the proof of the Marsden-Ratiu theorem on Poisson reduction by distributions (J. E. Marsden and T. S. Ratiu. “Reduction of Poisson manifolds”. In: Letters in Mathematical Physics 11.2 (1986), pp. 161–169).
Context: let $(M,\{\cdot,\cdot\})$ be a Poisson manifold and denote by $\Pi$ the corresponding bivector field. Let $N\subseteq M$ be a submanifold with inclusion $i:N\to M$ and let $E\subseteq TM|_N$ be a subbundle. Assume that
- $E\cap TN$ is an integrable subbundle, defining a foliation $\mathscr{F}$ on $N$,
- the leaf space $N/\mathscr{F}$ is a smooth manifold such that the projection $\pi:N\to N/\mathscr{F}$ is a submersion,
- for any two $F,G\in C^\infty(M)$ such that $dF(E)=dG(E)=0$ we have that $d\{F,G\}(E)=0$.
We say that the triple $(M,N,E)$ is Poisson reducible if there is a Poisson bracket $\{\cdot,\cdot\}'$ on the leaf space such that for any two $f,g\in C^\infty(N/\mathscr{F})$, if $F,G\in C^\infty(M)$ are extensions of $\pi^*f$ and $\pi^*g$, respectively, such that $dF(E)=dG(E)=0$, then $\pi^*\{f,g\}'=i^*\{F,G\}$.
The Marsden-Ratiu theorem states that a triple $(M,N,E)$ is Poisson reducible if and only if $\Pi^\sharp(E^0)\subseteq E+TN$, where $\Pi^\sharp:T^*M\to TM$ is the bundle homomorphism associated to the bivector field $\Pi$ and $E^0\subseteq T^*M$ is the annihilator of $E$.
The proof is not difficult, but there's one detail I don't understand. At the beginning of the proof they fix $x\in M$, $\beta\in E^0_x$ and $\alpha\in(E+TN)^0_x$ and they assert that there are functions $F,K\in C^\infty(U)$ in some neighborhood $U$ of $x$ such that $dF(E)=dK(E)=0$, $dF_x=\beta$, $dK_x=\alpha$ and $K|_N\equiv 0$. The rest of the proof is easy to follow, but I don't see why these two extensions always exist. The argument shouln't be too complicated since they think it obvious, but I just don't see it.