I have problem how to obtain symplectic $4\times 4$ matrix $T$ with one more condition. Matrix $H$ is known and I have it in analytical form, but the problem is how to obtain matrix $T$ which is not unique and according to that, I need to find one which satisfy second condition
symplectic condition: $T^T J T=J$
additional: $T^T H T=\begin{pmatrix} \lambda & 0 \\ 0 & \lambda \\ \end{pmatrix}_{4*4 }, $
where
$$J=\begin{pmatrix} 0 & I \\ -I & 0 \\ \end{pmatrix},\qquad H=\begin{pmatrix} H_1 & H_2 \\ H_3 & H_4 \\ \end{pmatrix}_{4*4 },\qquad \lambda =\begin{pmatrix} \lambda _1 & 0 \\ 0 & \lambda _2 \\ \end{pmatrix}_{2*2 },\qquad I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}.$$
I started from condition 2) but system is complex. Any comment or suggestion what to do?