model corresponding to $F(\omega)=\omega$ [@kremer] (for a discussion of these models in connection to the freeze out stage in heavy ion collisions see [@linking; @progress]).

From Eq. (\[linea\]) we then have I\^[(0)]{}\_ = - R\[FR\[\]\] \[i0aux\] and I\^[a]{}\_ = - R\[FR\[\^[a]{}\]\] \[iaaux\] with R\[\] = - R\_p\^ and \_a R\[\_a\] = \_a \_a - R\^[bc]{} \_c \_b \[ra\]

The integrability conditions then read p\^I\^[(0)]{}\_ = 0 \[int0\] and I\^[a]{}\_ = 0 \[inta\] Note that there are no restrictions on $\left\langle p^\rho I^{a}_{\beta}  \right\rangle$ or on $\left\langle I^{(0)}_{\beta}  \right\rangle$.

### Variational equations

In our case the conserved currents are $J^{a\mu}$ and $T^{\mu\nu}$, so that the variational problem becomes \[S\^\_[;]{} - \_T\^-\_[a]{}J\^[a]{}\] = 0 where $\lambda_{\mu\nu}$ and $\lambda_{a\mu}$ are Lagrange multipliers forcing the energy momentum tensor and the color currents to take on their known values.

We then have

&= \_ + \_[a]{}\
&= \_ + \_[a]{}

\[varia\]

Using Eqs. (\[shearfull\]), (\[jdfull\]) and (\[entaux\]), together with Eqs. (\[i0aux\]) and (\[iaaux\]), the variational equations (\[varia\]) become R\[FR\[\]\] = \_ p\^p\^+ g\_\^a p\^\[variat1\] and R\[FR\[\^[b]{}\]\] = \_ p\^p\^\^b + g\_\^a p\^\[variat2\]

We will solve the equations (\[variat1\]) and (\[variat2\]) to second order in the relaxation time $\tau$. To this end, we will follow [@linking] 