The field equations for the chiral invariant model of pion dynamics developed by Charap have been revisited. Two new types of solutions of these equations have been obtained. Each type allows infinite number of solutions. It has also been shown that the chiral invariant field equations admit invariance for a transformation of the dependent variables.

Two sets of nonlinear partial differential equations originating from two different physical situations have been combined and a new set of nonlinear partial differential equations has been formed wherefrom the previous two sets can be obtained as particular cases. One of the two sets of equations was obtained by Yang [1] while discussing the condition of self-duality ofSU(2) gauge fields on Euclidean four-dimensional space. The second one was reported by Charap [2] for the chiral invariant model of pion dynamics under tangential parametrization. Using the same type of ansatz in each case De and Ray [16] and Ray [7] obtained physical solutions of the two sets of equations. Here exact solutions of the combined set of equations with particular values of the coupling constants have been obtained for a similar ansatz. These solutions too are physical in nature.

Some previously obtained physical solutions [1–3] of Yang’s equations forSU(2) gauge fields [4], Charap’s equations for pion dynamics [5,6] and their combination as proposed by Chakraborty and Chanda [1] have been presented. They represent different physical characteristics, e.g. spreading wave with solitary profile which tends to zero as time tends to infinity, spreading wave packets, solitary wave with oscillatory profile, localised wave with solitary profile which becomes plane wave periodically, and, wave packets which are oscillatory in nature.

It has been shown that the field equations for Charap’s chiral invariant model of the pion dynamics pass the Painlevé test for complete integrability in the sense of Weisset al. The truncation procedure of the same analysis leads to auto-Backlund transformation between two pairs of solutions. With the help of this transformation non-trivial exact solutions have been rediscovered

Painlevé test (Jimboet al [1]) for integrability for the Yang’s self-dual equations forSU(2) gauge fields has been revisited. Jimboet al analysed the complex form of the equations with a rather restricted form of singularity manifold. They did not discuss exact solutions in that context. Here the analysis has been done starting from the real form of the same equations and keeping the singularity manifold completely general in nature. It has been found that the equations, in real form, pass the Painlevé test for integrability. The truncation procedure of the same analysis leads to non-trivial exact solutions obtained previously and auto-Backlund transformation between two pairs of those solutions

Painlevé test for integrability for the combined equations generated from Yang's self-dual equations for $SU (2)$ gauge fields and Charap's equations for chiral invariant model of pion dynamics faces some peculiar situations that allow none of the stages (leading order analysis, resonance calculation and checking of the existence of the requisite number of arbitrary functions) to be conclusive. It is also revealed from a comparative study with the previous results that the existence of abnormal behaviour at any of the stated stages may have a correlation with the existence of chaotic property or some other properties that do not correspond to solitonic behaviour.