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.

The singular manifold method is used to generate lump solutions of a generalized integrable nonlinear Schrödinger equation in 2 + 1 dimensions. We present several essentially different types of lump solutions. The connection between this method and the Ablowitz-Villarroel scheme is also analysed.

We derive a class of localized solutions of a 2+1 nonlinear Schrödinger (NLS) equation and study their dynamical properties. The ensuing dynamics of these configurations is a superposition of a uniform, "center of mass" motion and a slower, individual motion; as a result, nontrivial scattering between humps may occur. Spectrally, these solutions correspond to the discrete spectrum of a certain associated operator, comprised of higher-order meromorphic eigenfunctions.

The Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the equation is increased.