Let $(M,\omega)$ be any symplectic manifold. The space of symplectic forms is open in the space of closed $2$-forms $\Omega(M)$, let $Symp(M)$ be the set of symplectic forms on $M$, the map $Symp(M)\rightarrow H^2(M,R)$ which sends $\omega$ to its $[\omega]$ is open. So in any neighborhood of a non trivial integral form there exists a symplectic form whose group of period is dense.

Let $(M,\omega)$ be any compact symplectic manifold. The space of symplectic forms is open in the space of closed $2$-forms $\Omega(M)$, let $Symp(M)$ be the set of symplectic forms on $M$, the map $Symp(M)\rightarrow H^2(M,R)$ which sends $\omega$ to its $[\omega]$ is open. So in any neighborhood of a non trivial integral form there exists a symplectic form whose group of period is dense.