I am trying to study algebraic properties of solutions of differential equations and hoping to get some guidance from the point of view of algebraic geometry.
Let $\frac{du^k}{dt}=F^k(u^1,...,u^n)$, $k=1,...,n$, be an autonomous system of differential equations, where $F^k: \mathbb{C^n}\rightarrow \mathbb{C^n} $ are sufficiently smooth functions that may have only isolated zeros and infinities.
Let $u^k=u^k\left(t,C^1,C^2,...,C^n\right)$, $k=1,...,n$, be a general solution of the differential system above, where $C^1,...,C^n$ are constants of integration.
Let $\phi^1(u)=t+C^1$, $\phi^k(u)=C^k$, $k=2,...,n$.
If $u_1, u_2$ are two solutions, let us define a binary operation $*$ on the set of solutions as follows: $\phi^k(u_1*u_2)=\ln{\left[e^{\phi^k(u_1)}+e^{\phi^k(u_2)}\right]}+Q^k\left(\phi^1(u_2)-\phi^1(u_1),...,\phi^n(u_2)-\phi^n(u_1)\right)$, where $Q^k$ is a polynomial.
Originally $Q^k$ is an arbitrary function. Here I am hoping that if it is a polynomial in $k$ variables over $\mathbb{C}$, this will attract attention of algebraic geometers.
I found that fixed solutions (or stationary points) of the system form a closed set under the binary operation $*$. Same can be said about periodic solutions.
Also, if $Q$ is even, then the operation $*$ is commutative, and if $Q\equiv0$, it is associative.
I want to find more properties of the operation $*$.