Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways. In this paper, taking the well known Kadomtsev–Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in (1+1)-dimensions, (2+1)-dimensions, (3+1)-dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

A new procedure is firstly proposed to construct soliton equations with self-consistent sources (SESCSs) in bilinear forms, starting from the Gram-type determinant solution or Gram-type Pfaffian solution of soliton equations without sources; the soliton solutions of SESCSs can then be given. This procedure is applied to the 2D Toda lattice equation, the discrete KP equation and the BKP equation.

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