content_image/340569.jpg content_image/340577.jpg content_image/340576.jpg content_image/340575.jpg
<img src='content_image/340569.jpg'>

terms of the time variable T (Fig.1). Again, We have recovered a monotonic dependence of our new variable T with respect to the isotropic variable of the Universe a and the universe expansion begins with no singularity. 

## Isotropic Bianchi type IX : 

In particular, we are more interested to find the behavior of the isotropic variables a and p$_{a}$ when β$_{+}$ = 0 and β − = 0. We find the equations of motions as, 

and 

<img src='content_image/340577.jpg'>

<img src='content_image/340576.jpg'>

The nontrivial solution of the above coupled non linear differential equations (38) and (39), for the scale factor gives 

<img src='content_image/340575.jpg'>

where c$_{1}$ and c$_{2}$ are positive constants of integrations. This solution shows that the universe has no singularity at all. Besides, the universe shrinks to a big crunch as T goes to infinity while it reaches a maximum size during its history. Note that, in the isotropic limit, the Bianchi IX model contains the closed FRW model and the behavior of the scale factor a is as expected for this model. 

