On the density of solutions of an equation in $\mathbf{CP}^2$

In this paper we consider the system
\begin{equation} \dot u=P(u), \end{equation}
where $u=(u_0,u_1,u_2)\in\mathbf C^3$, $P=(P_0,P_1,P_2)$ and the $P_i$ are homogeneous polynomials of degree $2n$ ($n\geqslant1$) with complex coefficients. Let $A_n$ be the space of coefficients of the right-hand sides of the system (1). Any point $\alpha\in A_n$ defines a system of the form (1).
Our aim in this paper is to show that the property of the solutions of the system (1) being dense in $\mathbf{CP}^2$ is locally characteristic, i.e. we prove that in $A_n$ there exists an open set $U$ such that the solutions of the system (1) with right-hand side $\alpha\in U$ are everywhere dense in $\mathbf{CP}^2$.
This result can be extended without difficulty to the case in which the degree of the homogeneous polynomials appearing in the right-hand side of the system (1) is odd.
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