We use the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair.

The Painlevé analysis is a simple and reliable tool for testing the integrability of nonlinear ODEs and PDEs [

The very first step of the Weiss-Kruskal algorithm, however, has no counterpart in the Ablowitz-Ramani-Segur algorithm: starting with the Painlevé analysis of a nonlinear PDE, one must determine which of one analytic hypersurfaces is characteristic for the tested equation, in order to perform the whole subsequent analysis of solutions around noncharacteristic hypersurfaces only. Ward [

In the present paper, in Section

This nonlinear PDE (

Section

Let us take the fourth-order three-dimensional nonlinear PDE (

It is an easy task to perform the Painlevé analysis of (

The PDE (

At first sight, such a complicated branching of solutions of (

Let us proceed to Weiss’ objections [

Now, let us return to the Calogero equation (

Let us try to find a Bäcklund transformation of the Calogero equation (

First, we employ the method of truncated singular expansions of Weiss [

It looks strange, however, that the

We have obtained two auto-Bäcklund transformations for Calogero equation (

We have shown that the method of truncated singular expansions does not lead to the simplest auto-Bäcklund transformation of Calogero equation (

In this paper, we used the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs.

First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Of course, if the Painlevé property is considered as an abstract analytic property, one may give any definition of it. However, if the Painlevé property is defined to be used as an indicator of integrability of nonlinear equations, the adequacy of its definition becomes an experimental result. By the singularity analysis of the Calogero equation, we have shown that Ward’s definition of the Painlevé property for PDEs is well founded.

Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair. We have found two different Bäcklund transformations of the Calogero equation into itself: one follows from the truncated singular expansion, the other one follows from the Lax pair, and the former turns out to be a special case of the square of the latter. In other words, the way from the truncated singular expansions to Bäcklund transformations and Lax pairs is not so straightforward as it is sometimes stated in the literature.