§32.9 Other Elementary Solutions

Elementary nonrational solutions of ${\text{P}}_{\text{III}}$ are

with $\kappa$, $\lambda$, $\mu$, and $\nu$ arbitrary constants.

In the case $\gamma =0$ and $\alpha \delta \ne 0$ we assume, as in §32.2(ii), $\alpha =1$ and $\delta =-1$. Then ${\text{P}}_{\text{III}}$ has algebraic solutions iff

with $n\in \mathbb{Z}$. These are rational solutions in $\zeta =z^{1/3}$ of the form

where $P_{n^2+1}(\zeta )$ and $Q_{n^2}(\zeta )$ are polynomials of degrees $n^2+1$ and $n^2$, respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when $\delta =0$ and $\beta \gamma \ne 0$.

${\text{P}}_{\text{III}}$ with $\beta =\delta =0$ has a first integral

with $C$ an arbitrary constant, which is solvable by quadrature. A similar result holds when $\alpha =\gamma =0$. ${\text{P}}_{\text{III}}$ with $\alpha =\beta =\gamma =\delta =0$, has the general solution $w(z)=C{z}^{\mu }$, with $C$ and $\mu$ arbitrary constants.

Elementary nonrational solutions of ${\text{P}}_{\text{V}}$ are

with $\kappa$ and $\mu$ arbitrary constants.

${\text{P}}_{\text{V}}$, with $\delta =0$, has algebraic solutions if either

or

with $n\in \mathbb{Z}$ and $\mu$ arbitrary. These are rational solutions in $\zeta =z^{1/2}$ of the form

where $P_{n^2-n+1}(\zeta )$ and $Q_{n^2-n}(\zeta )$ are polynomials of degrees $n^2-n+1$ and $n^2-n$, respectively, with no common zeros.

${\text{P}}_{\text{V}}$, with $\gamma =\delta =0$, has a first integral

with $C$ an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). ${\text{P}}_{\text{V}}$, with $\alpha =\beta =0$ and ${\gamma }^2+2\delta =0$, has solutions $w(z)=C\exp \left(\pm \sqrt{-2\delta }z\right)$, with $C$ an arbitrary constant.

An elementary algebraic solution of ${\text{P}}_{\text{VI}}$ is

with $\kappa$ and $\mu$ arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of ${\text{P}}_{\text{VI}}$ with $\beta =\gamma =0$, $\delta =\frac{1}{2}$. 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).