In the book "Quantum Gravity in 2+1 dimensions" by Steven Carlip he writes down a possible modification to the Einstein-Hilbert Action in 3d (eq. 1.16 to eq. 1.18)
\begin{equation} I_{GCS}=-\frac{1}{32\pi G\mu}\int d^3x \epsilon^{\lambda \mu \nu}\Gamma ^{\rho}_{\lambda \sigma }\Big[\partial_{\mu}\Gamma^{\sigma}_{\rho \nu}+\frac{2}{3} \Gamma_{\mu\tau}^{\sigma}\Gamma_{\nu\rho}^{\tau}\Big]. \end{equation}
Without much explanation he says that the equations of motion for this action (plus the usual Einstein-Hilbert term) are
\begin{equation} G^{\mu \nu}+\frac{1}{\mu}C^{\mu \nu}=0 \end{equation}
where
\begin{equation} C_{\mu \nu}=\frac{1}{\sqrt{-g}}\epsilon^{\mu\rho\sigma}\nabla_{\rho}\Big[R^{\nu}_{\sigma}-\frac{1}{4}\delta^{\nu}_{\sigma}R\Big] \end{equation}
Where $R^{\nu}_{\sigma}$ is the Ricci tensor and $R$ is the curvature scalar.
My question is: is there any trick to solve this variation of the action? the math seems inhumanly hard and I'm not seeing any easy way this could be done.
