  The dynamics of the tilted axisymmetric Bianchi IX cosmological models are
explored allowing energy flux in the source fluid. The Einstein equations and
the continuity equation are presented treating the equation of state $w$ and
the tilt angle of the fluid $\lambda$ as time dependent functions, but when
analysing the phase space $w$ and $\lambda$ are considered free parameters and
the shear, the vorticity and the curvature of the spacetime span a
three-dimensional phase space that contains seven fixed points. One of them is
an attractor that inflates the universe anisotropically, thus providing a
counter example to the cosmic no-hair conjecture. Also, examples of a realistic
though fine-tuned cosmologies are presented wherein the rotation can grow
significant towards the present epoch but the shear stays within the
observational bounds. The examples suggest that the model used here can explain
the parity-violating anomalies of the cosmic microwave background. The result
significantly differs from an earlier study, where a non-axisymmetric Bianchi
IX type with a tilted perfect dust source was found to induce too much shear
for observationally significant vorticity.
