On the integrability of a generalized Davey–Stewartson system

• The method of Zakharov and Shulman is applicable to a three-component system.
• For a class of generalized Davey–Stewartson system necessary conditions for integrability are obtained.
• This class is shown to be not integrable unless it reduces to the known integrable cases.

In this work we investigate the integrable cases of the elliptic–hyperbolic–hyperbolic generalized Davey–Stewartson system introduced in Babaoğlu and Erbay (2004)  [6] following the method of Zakharov and Shulman (1980)  [3]. This method provides us with a set of algebraic conditions on the parameters of the system, which are just necessary conditions for the system to be integrable by means of the inverse scattering transform. Taking into account the constraints arising from the physical derivation of the generalized Davey–Stewartson system as described in Babaoğlu and Erbay (2004)  [6], we show that this system is integrable only when it can be transformed to an integrable case of the Davey–Stewartson system.