Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$. We call such an embedding isosymplectic.

The group $\mathrm{Symp}(M, \omega)$ of symplectomorphisms of $M$ acts on $\mathcal{E}$ from the right by composition of mappings and this action is free and continuous in the compact-open $C^{\infty}$-topology. Therefore we get a projection $$ \mathrm{Symp}(M, \omega) \to \mathcal{E} \xrightarrow{p} \mathcal{E}\,/\, \mathrm{Symp}(M, \omega) =:\mathcal{B}.$$

Question 1: Is $p\colon \mathcal{E} \to \mathcal{B}$ a locally trivial fibre bundle?

Question 2: Have there been any attempts to show that $\mathcal{E}$ is an infinite dimensional smooth manifold modeled on convenient locally convex spaces or Fréchet spaces?

I know that the space $\mathrm{Emb}(M, N)$ of all embeddings of $M$ into $N$ is a convenient infinite dimensional manifold (Kriegl, Michor [1]). So is the group $\mathrm{Symp}(M, \omega)$, but here to find local charts is not as easy as one would (maybe) expect, so it is probably even harder to find local charts on $\mathcal{E}$.

[1] A. Kriegl, P. W. Michor: The convenient setting of global analysis.