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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 $\omega_{1}$...

Background van Holten's algorithm (see e.g. here and here) is a way of constructing or recognizing dynamical/hidden symmetries in classical mechanics by looking for Killing tensors on the ...