This question is motivated by the classical fact from differential geometry : 


*Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ acts transtively on the configuration space of $n$-points in $M$ or equivalently it acts $n$-transitively on $M$.*


As I recall, it is known that the symplectomorphism group $(M,\omega)$ acts transitively on $M$, which is assumed to be symplectic. My question is then the following :


*Let $(M,\omega)$ be a symplectic manifold.  

(i) When does $\textrm{Symp}(M,\omega)$ act $n$-transtively for $n\geq 2$ ?  

(ii) If the answer above is NOT ALWAYS then what is known ?*


As some background, the usual way one proves (rather the only way I know how to prove this) the first fact is by showing the following :  

(i) for two sets of distinct $n$ points in $M$ given by $\{p\_1,\ldots,p\_n\}$ and $\{q\_1,\ldots,q\_n\}$ which are close, we find disjoint disks $D\_i$'s containing $p\_i,q\_i$. This requires dimension at least $2$. Use some diffeomorphism of $D\_i$ that is smoothly identity at the boundary and looks like a rotation inside $D\_i$ that swaps $p\_i$ and $q\_i$.  

(ii) Define the natural equivalence relation on $n$-tuples and observe that the configuration space of $n$-points in $M$. By (ii) each equivalence class is open. It is alsoclosed being the complement of open sets. Since the configuration space is connected (this requires dimension at least $2$) this means there is only one equivalence class. 


Does this idea work in the symplectic setting - perhaps by taking paths $\gamma\_i$ from $p\_i$ to $q\_i$ and getting Hamiltonian vector fields via $\omega(\gamma\_i',\cdot)$ ?



How transitive are the actions of symplectomorphism groups ?
Dear Somnath Basu, the answer to question is that, for any $k\geq 2$, the k-fold transitivity of the action of $\mathrm{Sympl}(M,\omega)$ on $M$ has only one obstruction, the trivial one, i.e. connectivity of $M$.  

But has you proposed there is even more.


In particular [Theorem A in a paper of W. Boothby](http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0236961-0/S0002-9947-1969-0236961-0.pdf) says that, 



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> given a connected symplectic manifold $(M,\omega)$, for any two sets $\{x\_1,\ldots,x\_n\}$ and $\{y\_1,\ldots,y\_n\}$ of disjoint point in $M$, where $n$ is arbitrary natural number, there exists a time dependent hamiltonian vector field $X\_t$ of $(M,\omega)$ such that its evolution operator $K^X\_{1,0}$ maps $x\_i$ to $y\_i$ for $i=1,\ldots,n$.
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