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Sep 27, 2020 at 15:44 comment added Francois Ziegler @DorianoBrogioli I don’t know! Your modified question seems hard to make precise.
Sep 27, 2020 at 15:02 comment added Doriano Brogioli Rigorously, this is the answer! However, the aim of my question was slightly different. I was wondering: I have an $H$, and a $K$ in involution, but the flux of $K$ is not proper so I cannot apply M-W. I believed that this meant that I could not reduce $H$. But here, instead, $H$ is already "reduced", in the sense that it does not depend on one of the variables (actually, it does not depend on two, $z_1$ and $z_2$). Is this a general property, i.e. that, whenever $K$ is not proper, there is such a reduction, in some extended sense? Do you suggest me to open a new question?
Sep 27, 2020 at 14:47 vote accept Doriano Brogioli
Sep 27, 2020 at 13:39 history edited Francois Ziegler CC BY-SA 4.0
added 39 characters in body
Sep 27, 2020 at 13:22 history answered Francois Ziegler CC BY-SA 4.0