On the second Painlevé equation and its higher analogues

Six Painlevé equations were obtained by Paul Painlevé and his school during the classification of ODE's of the form $w'' = P (z, w, w')$, where the function $P (z, w, w')$ is a polynomial in $w$ and $w'$ and is an analytic function of $z$. These equations are widely used in physics and have beautiful mathematical structures. My talk is devoted to the second Painlevé equation.
We will discuss the integrability of this equation and introduce its Hamiltonian representation in terms of the Kazuo Okamoto variables. On the other hand, the PII equation is integrable in the sense of the Lax pair and the isomonodromic representation, that I will present.
The Bäcklund transformation and the affine Weyl group are another interesting question. Using these symmetries, we are able to construct various rational solutions for the integer parameter PII equation.
The second Painlevé equation has one more important representation in terms of $\sigma$-coordinates which are $log$-symplectic.
There are higher analogues of the PII equation, which we will obtain by self-similar reduction of the modified Korteveg-de Vries hierarchy.