Yeah that would be nice

I'm not sure which convolution operation you mean, but when one is passing to the limit $\epsilon \to 0$, the evolution equation is in general non linear, e.g. when one is considering the Schrödinger equation with a harmonic oszillator potential instead

The single time derivative simply comes from expressing the KGE as a first order system of equations. The last equation come from manipulating the expressions defined above and computing the eigenvalues of the associated symbol of the differential operator. I would like to understand the limits of the KGE in the framework of pseudo differential operators, therefore I chose the approach above

That is nice to know, but can you prove that rotations act transitive on the solutions?

I mean that $\eta = \frac{pi}{2}$ is a valid solution and I'm fine with it. But what's tells you that there isn't any other solution for $\eta$, where it isn't constant