Quantum gravity in affine picture

I am studying an article which is about quantum gravity (M. Martellini, "Quantum Gravity in the Eddington Purely Affine Picture," Phys. Rev. D 29 (1984) 2746). I have come to eq. 2.11a which is

begin{equation}
overline{nabla}_{rho}overline{K}_{alphabeta}=0
end{equation}

where, $K_{munu}$ is Ricci tensor

begin{align}
&Gamma^{alpha}_{munu}=overline{Gamma}^{alpha}_{munu}+sqrt{lambda}gamma^{alpha}_{munu}
&overline{Gamma}^{alpha}_{munu}=overline{Gamma}^{alpha}_{(munu)}, gamma^{alpha}_{munu}=gamma^{alpha}_{(munu)},
&overline{nabla}_{rho}(overline{Gamma})overline{K}(overline{Gamma})=0.
end{align}

Somehow, from the above equation of motion we should derive the eq. 2.11b which is

begin{align}
&[overrightarrow{overline{nabla}}_{rho}overrightarrow{overline{partial}}_{kappa}mathfrak{I}^{kappamunu}_{lambdaalphabeta}-2overline{K}_{lambda(alpha}delta^{munu}_{beta)rho}+(overline{nabla}_{rho}overline{Gamma}^{sigma}_{kappalambda})mathfrak{I}^{kappamunu}_{sigmaalphabeta}+(overline{nabla}_{rho}overline{Gamma}^{(mu}_{taueta})mathfrak{I}^{nu)taueta}_{lambdaalphabeta}+overline{Gamma}^{sigma}_{kappalambda}overline{nabla}_{rho}mathfrak{I}^{kappamunu}_{sigmaalphabeta}+overline{Gamma}^{(mu}_{taueta}overline{nabla}_{rho}mathfrak{I}^{nu)etatau}_{lambdaalphabeta}]gamma^{lambda}_{munu}
&=-2sqrt{lambda}gamma^{lambda}_{kappasigma}(overline{nabla}_{rho}mathfrak{I}^{kappamunu}_{lambdaalphabeta})gamma^{sigma}_{munu}+2sqrt{lambda}gamma^{delta}_{rhotau}[delta^{etatau}_{alphabeta}(partial_{kappa}mathfrak{I}^{kappamunu}_{lambdadeltaeta}+overline{Gamma}^{sigma}_{kappalambda}mathfrak{I}^{kappamunu}_{sigmadeltaeta}+overline{Gamma}^{(mu}_{kappasigma}mathfrak{I}^{nu)kappasigma}_{lambdadeltaeta})]gamma^{lambda}_{munu}+2lambdagamma^{delta}_{rhotau}(mathfrak{I}^{kappamunu}_{lambdadeltaeta}delta^{etatau}_{alphabeta})gamma^{lambda}_{kappasigma}gamma^{sigma}_{munu}
end{align}

where
begin{align}
&mathfrak{I}^{kappamunu}_{lambdaetadelta} equiv (delta^{mu}_{lambda}delta^{nukappa}_{etadelta}-delta^{kappa}_{lambda}delta^{munu}_{etadelta})
&delta^{nukappa}_{etadelta}equivfrac{1}{2}(delta^{nu}_{eta}delta^{kappa}_{delta}+delta^{nu}_{delta}delta^{kappa}_{eta})
end{align}

I have tried it in both ways:

1. Begining from covariant derivative of Ricci Tensor to find above eq.
2. Begining from the result to find covariant derivative of Ricci Tensor.

I could not find it in both ways. Can anyone help me? Besides, why are we using the equation of motion in this form?

Note: since we are in purely affine picture, we are not allowed to use the metric tensor.