Conditionally integrable PT -symmetric hierarchies and near-Lax pairs of the KdV and KPII equations

Earlier work is generalized to obtain novel conditionally-integrable hierarchies of various <?CDATA P T?> P T -symmetric, nonlinear partial differential equations. The Painlevé Test yields the possible integrable cases. These are labeled by the integer n , the order of the dominant pole in the Laurent expansion for the solution. For the <?CDATA P T?> P T -symmetric Korteweg–de Vries (KdV) equation which has been considered earlier, further analysis using the extended homogeneous balance technique gives the near-Lax Pair. Next, a <?CDATA P T?> P T -Symmetric hierarchy of (2 + 1) Kadomtsev-Petviashvili equations is considered. The Painlevé Test and Invariant Painlevé analysis in (2 + 1) dimensions yield special solutions and Bäcklund Transformations for the integrable cases.