  The Szekeres system with cosmological constant term describes the evolution
of the kinematic quantities for Einstein field equations in $\mathbb{R}^4$. In
this study, we investigate the behavior of trajectories in the presence of
cosmological constant. It has been shown that the Szekeres system is a
Hamiltonian dynamical system. It admits at least two conservation laws, $h$ and
$I_{0}$ which indicate the integrability of the Hamiltonian system. We solve
the Hamilton-Jacobi equation, and we reduce the Szekeres system from
$\mathbb{R}^4$ to an equivalent system defined in $\mathbb{R}^2$. Global
dynamics are studied where we find that there exists an attractor in the finite
regime only for positive valued cosmological constant and $I_0<2.08$.
Otherwise, trajectories reach infinity. For $I_ {0}>0$ the origin of
trajectories in $\mathbb{R}^2$ is also at infinity. Finally, we investigate the
evolution of physical properties by using dimensionless variables different
from that of Hubble-normalization conducing to a dynamical system in
$\mathbb{R}^5$. We see that the attractor at the finite regime in
$\mathbb{R}^5$ is related with the de Sitter universe for a positive
cosmological constant.
