I have a lattice in $\mathbb R^{2n}$ specified by some basis M (with each row of M being a basis vector), and I am interested in the properties of $A = MJM^T$, where $J = \begin{pmatrix}0 & I_n\\ -I_n & 0 \end{pmatrix}$ is the standard symplectic form. In particular, my question is the following: what are the conditions under which there exists an orthogonal transformation $O \in \mathrm O(\mathbb R,2n)$ of the basis vectors of $M$, $M' = MO$, such that $A' = M'J(M')^T = MOJO^TM^T$ contains only integers ? Even better, is there a procedure to find such $O$ ?
Equivalently, what are the conditions under which there exists a bilinear form $J' = OJO^T$ such that $A'$ is integral...
One necessary condition is that $\det(M) \in \mathbb Z$. Since we impose that $\det A' \in \mathbb Z$ and $A'$ is skew-symmetric by construction, we have $\det A' = \mathrm{Pf}(A')^2$, $\mathrm{Pf}(A') \in \mathbb Z$ and $\det(A') = \det(M')^2$, such that $\det(M) \in \mathbb Z$. $\mathrm{Pf}(A)$ is the Pfaffian of A.
Also, it is clear that if such a $O$ exists, it is not unique since if $O$ is a solution, then $O' = OS$ is a solution as well for all symplectic transformations $S \in O(\mathbb R,2n)\cap \mathrm{Sp}(\mathbb R,2n)$.
Thank you for any help regarding this problem!