I'm mostly interested in two dimensional systems $$\dot{x}=f(x,y), \\ \dot{y}=g(x,y), $$ which include the second order ODEs $$\ddot{x}=F(t,x,\dot{x}). $$ Putting the linear equations aside, on which we have a comprehensive literature, what are (some of) the nonlinear ODEs for which we have a formula for the general solution, and preferably some understanding of its behaviour? An example is the Weierstraß differential equation $$\ddot{x}=6x^2-\frac{1}{2}g_2 $$ for which the solution is given by $$x(t)=\wp \left(t-t_0;g_2,g_3\right) $$ where $t_0$ and $g_3$ are the integration constants, and $\wp$ is the Weierstraß elliptic function.
Thanks!
EDIT: I'm interested in as many examples as possible, but partial answers are also greatly appreciated.