The first Painlevé equation
$$P\_I:y''=6y^2-x $$
has the symmetries $$x \mapsto \omega x \\y \mapsto \omega^3 y$$
for any fifth root of unity $\omega$. At the same time, the near-infinity asymptotics of $P\_I$ involve the five rays $$\Gamma\_k= \left\{x:\arg x= \frac{2 \pi i k}{5} \right\}, \quad k=0,1,2,3,4.$$


Looking at the second Painlevé equation
$$P\_{II}: y''=2y^3+xy+\alpha $$
the scaling symmetries are
$$x \mapsto \omega x \\ y \mapsto \omega^2 x $$
for any third root of unity $\omega$. However, the near-infinity asymptotics of $P\_{II}$ actually involve *six* rays
$$\Gamma\_k= \left\{x:\arg x= \frac{2 \pi i k}{6} \right\}, \quad k=0,1,2,3,4,5.$$
My question is: is there a direct relationship between the discrete symmetries of the second Painlevé equation, and its near-infinity asymptotics, as in the case of the first Painlevé equation? If so, how come I only found three symmetries, which should correspond to the rays
$$\left\{x: \arg x=\frac{2 \pi i k}{3} \right\}, \quad k=0,1,2 $$


Thanks!



Relation between symmetries and asymptotics for Painlevé equations
Yes, there are relationships, they depend on the weights chosen to compactify the system at infinity. First, write the equation as the companion autonomous vector field in $\mathbb C^3$ $$X(x,y\_1,y\_2)=\partial\_x+y\_2\partial\_{y\_1}-6z\_1^2\partial\_{z\_2}$$
 For $P\_1$, you use the Boutroux compactification given by the chart transition mapping $(t,u\_1,u\_2)\mapsto(x,y\_1,y\_2)$ with $$\begin{cases}{x=t^{-4/5}}\\{y\_1=(u\_1+\frac{i}{\sqrt6})t^{-2/5}}\\{y\_2=u\_2t^{-3/5}}\end{cases}$$The new vector field, after division by $-\frac{5}{4t^{1/5}}$, writes $$W(t,u\_1,u\_2)=t^2\partial\_t+(\cdots)\partial\_{u\_1}+(\cdots)\partial\_{u\_2}$$which has an irregular singularity at $(0,0,0)$ with Poincaré rank-$1$. 


The lines $\Gamma\_k$ are the preimages in the starting chart of the Stokes line of the previous vector field. There are five of them because of the order-$5$ ramification in the $x$-variable. 


In the case of $P\_2$ after compactification and reduction, you obtain two irregular singularities at infinity (Poincaré rank-$1$), each one bringing a Stokes line in. Hence the $6=2\times3$ lines in the original chart. (I need to check that precisely, though, I'll update the answer).


