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!