The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota τ -function method. In this paper, we report how to improve the singular manifold method when the equation has more than one Painlevé branch. The singular manifold method generalized in such a way is applied to a pair of equations in 2 + 1 dimensions

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a Bäcklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.

A generalization of the negative Camassa-Holm hierarchy to 2 + 1 dimensions is presented under the name CHH(2+1). Several hodograph transformations are applied in order to transform the hierarchy into a system of coupled CBS (Calogero-Bogoyavlenskii-Schiff) equations in 2 + 1 dimensions that pass the Painlevé test. A non-isospectral Lax pair for CHH(2+1) is obtained through the above mentioned relationship with the CBS spectral problem.

In this paper the nonlinear equation mly = ( m , + mxmy)x is throughly analysed.The Painlev6 test is performed yielding a positive result. The Bgcklund transformations are found and the Darboux-Moutan-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painlev6 analysis, is proved to be a useful tool for genenring solutions. Some interesting explicit expressions for one and W O solitons are obtained and analysed in such a way. This system may be considered as a model for an incompressible fluid where U and U are the components of the velocity. The spectral transform for this system has been investigated in [l] and more details can be found in [2]. This system has been considered in [3] as a generalization to 2 + 1 of the results from Hirota and Satsuma [4]. The nonclassical symmetries, PainlevC property and solutions of the same 1 + 1 reduction of (1.1) considered in [4] have been studied by Clarkson and Mansfield [5]. It is also interesting to notice that for x = y the system (1.1) reduces to the KdV equation [l].

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painleve property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so-called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painleve property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painleve property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.