Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Picard and Chazy solutions to the Painleve' VI equation   I study the solutions of a particular family of Painlev\'e VI equations with
the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for
$2\mu\in\interi$. I show that the case of half-integer $\mu$ is integrable and
that the solutions are of two types: the so-called Picard solutions and the
so-called Chazy solutions. I give explicit formulae for them and completely
determine their asymptotic behaviour near the singular points $0,1,\infty$ and
their nonlinear monodromy. I study the structure of analytic continuation of
the solutions to the PVI$\mu$ equation for any $\mu$ such that
$2\mu\in\interi$. As an application, I classify all the algebraic solutions.
For $\mu$ half-integer, I show that they are in one to one correspondence with
regular polygons or star-polygons in the plane. For $\mu$ integer, I show that
all algebraic solutions belong to a one-parameter family of rational solutions.

35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an