it is possible to show that the system admits the following integrals of motion $$\label{integrale1}
\frac{\dot{R_{i}}}{R_{i}}=H+s_{i}R^{-3}e^{\varphi}.$$ Here $\dot{\psi}\equiv -2\chi\eta F$ and the quantities $s_i$ are such that $s_{1}+s_{2}+s_{3}=0$. The evolution of $H$ is obtained using the trace of the $ii$-components combined with the $00$-component (\[Einst4\]) of the Einstein equation so getting $$\label{H_punto}
\dot{H}=\chi\rho-\frac{1}{2}\chi h-3H^{2}+\frac{3}{2}\chi\zeta FH+\chi(\frac{1}{2}\zeta-\frac{1}{3}\eta)\dot{F},$$ where $$\label{enthalpy}
h=\rho+p$$ represents the specific enthalpy. Moreover, the hydrodynamic equations $\nabla_{\nu}T_{\mu}^{\nu}=0$ (only the $0$-component is not vanishing here) provide the following evolution for $\rho$ $$\label{rho_punto}
  \dot{\rho}=-4\chi\eta F\rho-3Hh+9H^{2}\varsigma F+12H^{2}\eta F+3H\varsigma\dot{F}-2H\eta\dot{F}.$$ The first integrals (\[integrale1\]) can be re-cast in a more compact form by the use of (\[H\_punto\]) and (\[Einst4\]) as $$\label{integrale2}
\rho=\frac{1}{\chi}\left(3H^{2}-q^{2}R^{-6}e^{2\varphi}\right),$$ where $q^{2}\equiv \frac{1}{2}\left(s_{1}^{2}+s_{2}^{2}+s_{3}^{2}\right)$. It worth noting that setting $q^2=0$ corresponds to the isotropic case. Equations (\[H\]), (\[H\_punto\]), (\[rho\_punto\]) and (\[integrale2\]) together with the $ii$-components of the field equations represent the full set of the dynamical equations characterizing the present model. In order to close the system we introduce a polytropic index $\gamma$ and we consider an equation of state of the form $$\begin{aligned}
h=&\gamma\rho, && 1\leq\gamma\leq2\end{aligned}$$ where, e.g., $\gamma=1$ corresponds to dust matter and $\gamma=\frac{4}{3}$ to the radiation cases, respectively.

Asymptotic solutions with matter creation {#asymptotic-solutions-with-matter-creation .unnumbered}
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As well known [@BKL70], approaching 