Introduction

In [1] the Painlevé property for partial differential equations was defined. Briefly, we say that a partial differential equation has the Painlevé property when the solutions of the p.d.e, are ``single-valued'' about the movable, singularity manifold and the singularity manifold is ``noncharacteristic''. To be precise, if the singularity manifold is determined by

$$\varphi(z_1,z_2,\ldots,z_n)=0\eqno (1.1)$$

and $u=u(z_1,\ldots,z_n)$ is a solution of the p.d.e., then we require that

$$u=\varphi^{\alpha}\sum_{j=0}^{\infty} u_j\varphi^j\ ,\eqno (1.2)$$

where $u_0\ne 0$, $\varphi=\varphi(z_1,\ldots,z_n)$, $u_j=u_j(z_1, \ldots,z_n)$ are analytic functions of $(z_j)$ in a neighborhood of the manifold (1.1), and $\alpha$ is an integer. The requirement that the manifold (1.1) be noncharacteristic insures that the expansion (1.2) will be well defined, in the sense of the Cauchy-Kowalevsky theorem. Substitution of(l.2) into the p.d.e, determines the value(s) of $\alpha$, and defines the recursion relations for $u(j)$, $j=0,1,2, \ldots$ . When the anzatz (1.2) is correct, the p.d.e, is said to possess the Painlevé property and is conjectured to be integrable. The ``Painlevé conjecture'', as originally formulated by Ablowitz [2], states that when all the ordinary differential equations obtained by exact similarity transforms from a given partial differential equation have the Painlevé property, then the partial differential equation is ``integrable''. The above definition of the ``Painlevé property'' allows this conjecture to be stated directly for the partial differential equation.

In [3] Bäcklund transformations were obtained by truncating the expansion (1.2) at the ``constant'' level term. That is, we set

$$u=u_0\varphi^{^{-N}}+u_1\varphi^{^{-N+1}}+\cdots+u_{_N}\eqno (1.3)$$

and find, from the recursion relations for $u_j$, an overdetermined system of equations for $(\varphi,u_j,j=0,1,\ldots,N)$, where $u_{_N}$ will satisfy the (original) p.d.e. Upon solving the overdetermined system, it was found, for those equations considered, that $\varphi$ satisfied an equation formulated in terms of the Schwarzian derivative:

$$\{\varphi;x\}={\partial\over\partial x}\left({\varphi_{xx}\ov... ...r 2} \left({\varphi_{xx}\over\varphi_x}\right)^2\ .\eqno (1.4)$$

The invariance of (1.4) under the Moebius group

$$\varphi={a\psi+b\over c\psi+d}\ ,\qquad\{\varphi;x\}=\{\psi;x\}\eqno (1.5)$$

motivates the substitution

$$\varphi=v_1/v_2\ ,\eqno (1.6)$$

by which the Lax pairs may be found [3].

Investigation of a certain class of equations formulated in terms of the Schwarzian derivatives revealed that these equations have the Painlevé property about movable, singularity manifolds of order $-1$. However, the occurrence of an additional type of movable singularity prevents this class of equations from identically possessing the Painlevé property. Hence, nonintegrable behavior can arise [2].

In this paper a restriction (symmetry) is imposed that allows one to conclude that, when an equation is formulated in terms of the Schwarzian derivative and has this ``symmetry'', the equation identically possesses the Painlevé property. Within this class of equations are found the KdV, Caudrey-Dodd-Gibbon and Kuperschmidt equations. Furthermore, the ``symmetry'' property and invariance under the Moebius group allow effective Bäcklund transforms to be defined for these equations. In particular, rational or algebraic [in $(x,t)$] solutions can be generated iteratively.

In the next section, the Painlevé property and Bäcklund transformation for the KdV equation are reviewed for later reference.

In Sec. 3 the Painlevé property and Bäcklund transforms for the Caudrey-Dodd-Gibbon equation are presented. From these considerations the Kuperschmidt equation is found. The transformation between the Caudrey-Dodd-Gibbon and Kuperschmidt equations can be regarded as a certain ``symmetry'' under which these equations are ``dual''.

In Sec. 4, the ``symmetry'' discovered in Sec. 3 is employed to define a class of p.d.e.'s that possess the Painlevé property. The KdV equation is shown to be contained in this class of equations and self-dual w.r.t, this symmetry. Then, the sequences of higher order KdV, Caudrey-Dodd-Gibbon, and Kuperschmidt equations are found to be within this identically Painlevé class of equations and Bäcklund transformations are obtained for these sequences of equations.

In Sec. 5 rational [in $(x,t)$] solutions are constructed for several equations. In Appendix A the Lax pair for the Caudrey-Dodd-Gibbon equation is derived. In Appendix B further considerations relating to the seventh-order equations are presented.