Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $\mu=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $\mu \ll T\wedge\omega_{\mathbb{P}^2}$ implies $\lambda_2=\frac{1}{2} \log\ d$, where $\lambda_1 \geq \lambda_2$ are the Lyapunov exponents of $\mu$. Then, assuming $\mu\ll T\wedge\omega_{\mathbb{P}^2}$, we study slice measures $\nu :=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $\mu$ with respect to the trace measure $T\wedge\omega_{\mathbb{P}^2}$ ensuring $\mu=\nu$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.