Summary: This paper considers a variant of the Boussinesq system \[ \eta_t+((1+\alpha \eta)w)_x-\frac{\beta}6w_{xxx}=0,\;w_t+\alpha ww_x+\eta_x-\frac{\beta}2w_{xxt}=0, \] where \(\alpha\) and \(\beta\) are positive constants. A lot of approximate models like the Boussinesq system and the two-component Camassa-Holm system can be derived from this system. We study its travelling wave solutions and analyze its phase portraits by applying the qualitative analysis methods of planar autonomous systems. We obtain its solitary wave solutions, kink-like or antikink-like wave solutions, compacton-like wave solutions and periodic wave solutions. Some numerical simulations of solutions are also given.