This is a request for references. I'm not asking for the answer... Just give me a starting point! I'm interested in global linearization of linear equations. I guess the problem is well understood in math given its simplicity, but under "linearization" I only find local linearization (e.g. near equilibria), which I'm not interested in.

Consider a linear system of equations $$\dot{x} = L x $$ where $x$ is a vector. Now perform an invertible transformation $x \to y=f(x)$. In the new variables the system is in general nonlinear $$\dot{y} = f' L f^{-1} (y) $$ where $f'$ denotes the Jacobian matrix.

Now let's pose an inverse problem: Given a generic nonlinear system $$\dot{y}=F(y)$$ which general conditions should $F$ satisfy for there to exist a linear map $L$ and a transformation $f$ such that $F=f' L f^{-1}$? I would expect the answer to involve geometric concepts, like geodesic transport, parallelizability and the such...