  In this paper, we extend the basic model of the restricted four-body problem
introducing two bigger dominant primaries $m_1$ and $m_2$ as oblate spheroids
when masses of the two primary bodies ($m_2$ and $m_3$) are equal. The aim of
this study is to investigate the use of zero velocity surfaces and the
Poincar\'{e} surfaces of section to determine the possible allowed boundary
regions and the stability orbit of the equilibrium points. According to
different values of Jacobi constant $C$, we can determine boundary region where
the particle can move in possible permitted zones. The stability regions of the
equilibrium points expanded due to presence of oblateness coefficient and
various values of $C$, whereas for certain range of $t$ ($100 \le t \le 200$),
orbits form a shape of cote's spiral. For different values of oblateness
parameters $A_1~ (0<A_1<1)$ and $A_2~ (0<A_2<1)$, we obtain two collinear and
six non-collinear equilibrium points. The non-collinear equilibrium points are
stable when the mass parameter $\mu$ lies in the interval
($0.0190637,~0.647603$). However, basins of attraction are constructed with the
help of Newton Raphson method to demonstrate the convergence as well as
divergence region of the equilibrium points. The nature of basins of attraction
of the equilibrium points are less effected in presence and absence of
oblateness coefficients $A_1$ and $A_2$ respectively in the proposed model.
