We study a two-parameter family of real solutions of a special Painlevé equation of the third kind, $$ u-\frac{(u')^2}{u}-\frac{u'}{x}+4\frac{(n-1) u^2 -n}{x}+4 u^3 -\frac{4}{u}, $$ which is used in many models of mathematical physics. Using the method of isomonodromic deformations, we construct asymptotic formulae on the real semi-axis as $x\to\infty$, including the distribution of poles of the singular solution. For $n\gg 1$ we show that there are no real poles for $x<n/2$ and that to the right of the point $x=n/2$ the poles are distributed as zeros of Bessel functions. In a neighbourhood of this point we study the transition layer that matches the regular and singular solutions. It turns out that this transition layer extends to the complex plane of the variable $x$, and there are two types of lattices of poles outside and inside the circle $|x|=n/2$. Bibliography: 17 titles.

Let $G/K$ be an irreducible Hermitian symmetric spaces of compact type with the standard homogeneous complex structure. Then the real symplectic manifold $(T^*(G/K),\Omega)$ has the natural complex structure $J^-$. We construct all $G$-invariant K\"ahler structures $(J,\Omega)$ on homogeneous domains in $T^*(G/K)$ anticommuting with $J^-$. Each such a hypercomplex structure, together with a suitable metric, defines a hyperk\"ahler structure. As an application, we obtain a new proof of the Harish-Chandra and Moore theorem.