From theory of linear differential equations, it follows that all curves in projective plane CP2 ( and a CONE in CP3? ) is fully characterized by 3rd order general linear diff. equation with all projective transformations and argument substitutions. Invariants of these diff equations are related to invariants of corresponding projective space- anharmonic curves etc.

4 order general linear diif. equation is equivalent to all space curves in CP3.

But , we have to look backwards a little bit to find a structure of Projective line CP1 or even projective POINT CP0?

For structure of CP1 (complex projective line) , we have to study linear diff equations of order 2 . And their invariants. We will find ALL projectively invariant curves in CP1. Interestingly, some of them might be oscillating , or at least trying to ( if overdamped by viscosity term). The general form:

y'' + p1(x)*y'+ p2(x)*y = 0

For structure of CP0 (complex projective point) , we have to study first order linear diff equations. To misuse the terminology , we are looking for CURVES in complex projective point, real or imaginary.General form look like:

y'+ p(x) y = 0

But the solutions if p(x) is not constant can be many. So in that sense, CP0 has very complicated structure, potentially.

I wonder where can I find invariant theory of second and first order general homogeneous linear differential equation? Are there any invariants at all?

I did not find any invariants yet, seems no one else has, but I have understood that general first order diff equation describes a complex phase space per se, and solution to it will be giving relation between "speed" y' and "position" y so that the solution of this differential equation will be a curve in complex phase space and so the structure of CP0-equals the structure of such phase space.

The interesting thing about this phase space is the arbitrary complex function p(x) :

y'(x ) + p(x)* y (x) = 0

if we take e.g [m] as dimension for y, and time for x , y' will have dimension of speed [m/s] and p(x) - [1/s] or frequency, while its inverse- 1/(p(x) a dimension of time.

Obviously, any invariant of such diff equation will involve p(x) in some way since it is the the only coefficient in this equation not equal to 1-either a constant frequency or function with frequency dimension.

So, what are these invariants of general Complex phase space?