The propagator for the 2D heat equation in an arbitrary linear space is shown to give solutions of the two-component Kadomtsev - Petviashvilii (KP) equations, also called Davey - Stewartson system. This propagator is subject to the Klein - Gordon equation and its right-derivatives are required to be of rank one, that imply that it can be expressed in terms of solutions of the Dirac equation. Large families of solutions of the two-component Kadomtsev - Petviashvilii equations are constructed in terms of solutions of the heat and Dirac equations. Particular attention is paid to the real reductions of the Davey - Stewartson type, recovering in this way the line solitons and the multidromion solutions. Moreover, new solutions to the Davey - Stewartson I are presented as massive deformations of the dromion.