Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$ but is not completely integrable with respect to $\omega_2$

Note that we do not limit the symplectic structures to structures with constant coefficients.

I had already asked a weaker version of this question in the following link but I confess that I did not understand the details(How does the action of orthonormal group guarantee's the integrability?)

http://mathforum.org/kb/thread.jspa?forumID=253&threadID=1653483&messageID=5990478#5990478

**Correction of the above Mathforum link:** For $n=1$, the appropriate first integral showing complete integrability of $Det =xw-yz$ is $xz+yw$. It works on $M_{2}(\mathbb{R})\setminus \{0\}$. At the origin(the zero matrix) we loose the independence of differentials of two first integrals.