# §32.9 Other Elementary Solutions

## §32.9(i) Third Painlevé Equation

Elementary nonrational solutions of  are

32.9.1
32.9.3

with , , , and arbitrary constants.

In the case and we assume, as in §32.2(ii), and . Then  has algebraic solutions iff

32.9.4

with . These are rational solutions in of the form

32.9.5

where and are polynomials of degrees and , respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when and .

with has a first integral

with an arbitrary constant, which is solvable by quadrature. A similar result holds when .  with , has the general solution , with and arbitrary constants.

## §32.9(ii) Fifth Painlevé Equation

Elementary nonrational solutions of  are

32.9.7

with and arbitrary constants.

, with , has algebraic solutions if either

or

with and arbitrary. These are rational solutions in of the form

32.9.11

where and are polynomials of degrees and , respectively, with no common zeros.

, with , has a first integral

with an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). , with and , has solutions , with an arbitrary constant.

## §32.9(iii) Sixth Painlevé Equation

An elementary algebraic solution of  is

32.9.13

with and arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of  with , . For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b).