We study the qualitative behaviour of three classes of spatially homogeneous,
axially symmetric cosmological models containing perfect fluid and cosmological
constant. We consider space-times having the metric forms
ds^2=-c^2dt^2+a^2(t)dr^2+b^2(t)\left(\frac{dv^2}{1-kv^2} +v^2d\phi^2\right),
where the parameter $k$ gives the curvature of the 2-dimensional angular surfaces, and which can take the values +1, 0, $-1$, leading to the Kantowski-Sachs, Bianchi I and Bianchi III metrics, respectively.
If we assume that at a certain moment of time, which we can take as the present time $t_0$, the Hubble flux becomes approximately isotropized in the sense $(H_a)_0=(H_b)_0$, which restrict the space of solutions with the metrics given above, we conclude that all these solutions evolve towards isotropization except for Kantowski-Sachs wtih $\Omega_{k_{0}}>0$ and $\Omega_{\Lambda_{0}}<\Omega_{\Lambda_{l}}$, and for Bianchi III with $\Omega_{k_{0}}<0$ and $\Omega_{\Lambda_{0}}< \Omega_{\Lambda_{l}}$. Also, we conclude that Kantowski-Sachs and Bianchi III models don't quite reproduce a big bang scenario in the sense that the first one emerges from a cigar singularity and the second one emerges from a pancake singularity.