Recently, I asked a question on the Mathematica Stack Exchange website regarding the use of homogeneous coordinates in differential equations.
The question is about extending the interval of existence in differential equations using homogeneous coordinates. A prototypical example is the initial value problem for the tangent function. $$\begin{cases} y'=1+y^2, \\ y(0)=0, \end{cases}$$ with the corresponding interval of existence being $(-\pi/2,\pi/2)$.
Identifying $y$ with a coordinate on the projective line, via $y=y_1/y_2$, results in the underdetermined system $$\begin{cases} y_1'y_2-y_1y_2'=y_2^2 \left[1+ \left(\dfrac{y_1}{y_2} \right)^2 \right] ,\\ \dfrac{y_1(0)}{y_2(0)}=0. \end{cases} $$ Adding the normalization condition $y_1y_1'+y_2y_2'=0$ (which is a consequence of fixing $y_1^2+y_2^2$), and using the initial conditions $y_1(0)=0,y_2(0)=1$ results in a solvable system, with global solutions $y_1(t)=\sin(t),y_2(t)=\cos(t)$. Thus it can be said that the interval of existence now is the entire real line!
I was trying to see how far I can extend this method. In the link above, for instance, I showed that it works (at least numerically) for the $\sec$-$\tan$ system $$\begin{cases} u'=uv \\ v'=u^2, \\ u(0)=1, \\ v(0)=0. \end{cases} $$
However, with other systems such as the first Painlevé equation $$\begin{cases} u'=v, \\ v'=6u^2+t, \end{cases} $$ the numerical integrator appears to get stuck at the first pole.
My questions: Is there a way to employ homogeneous coordinates in order to extend the interval of existence in the case of the first Painlevé equation? More generally, what about systems $ \dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x})$ with the Painlevé property (where all spontaneous singularities are poles)? If so, I'd greatly appreciate some details. I'm not asking for the code here, just the appropriate mathematical formulation.
Thank you!