The Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation describes the nonlinear behaviour of long-wavelength weakly nonlinear ion-acoustic waves propagating obliquely to an external uniform (space independent) static (time independent) magnetic field in a plasma consisting of warm adiabatic ions and a superposition of two distinct population of electrons, one due to Cairns et al. (1995 Geophys. Res. Lett.22, 2709), which generates the fast energetic electrons, and the other the well-known Maxwell–Boltzman distributed electrons. It is found that the compressive or rarefactive nature of the ion-acoustic solitary wave solution of the KdV-ZK equation does not depend on the ion temperature if σc<0 or σc>1, where σc is a function of β1, nsc and σsc. This β1 is the non-thermal parameter associated with the non-thermal velocity distribution function of electrons (Cairns et al.), nsc is the ratio of the unperturbed number density of the isothermal electrons to that of the non-thermal electrons and σsc is the ratio of the average temperature of the non-thermal electrons to that of the isothermal electrons. The KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether σc<0 or σc>1. When 0 ≤ σc ≤ 1, the KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether σ>σc or σ<σc, where σ is the ratio of the average temperature of ions to the effective temperature of electrons. If σ takes the value σc with 0 ≤ σc ≤ 1, the coefficient of the nonlinear term of the KdV-ZK equation vanishes and for this case the nonlinear evolution equation of the ion-acoustic wave is a modified KdV-ZK (MKdV-ZK) equation. It is found that the four-dimensional parameter space, originated from the physically admissible values of the four-parameters β1, σ, σsc and nsc of the present extended plasma system, can be decomposed into five mutually disjoint subsets with respect to the critical values of the different parameters, and the nonlinear behaviour of the same ion acoustic wave in those subsets can be described by different modified KdV-ZK equations. A general method of perturbation of the dependent variables has been developed to obtain the different evolution equations. The applicability of the different evolution equations and their solitary wave solutions (along with the conditions for their existence) have been investigated analytically and graphically.