This question is related to the existence of Einstein metrics on tangent bundles where the metric is induced by the isotropic almost complex structures on the tangent bundle. I'm trying this on the hyperbolic space and I got the following equations as the necessary conditions for the existence of such metrics on $T\mathbb{H}^2$. We consider $(x_1,x_2,y_1,y_2)$ is coordinate system on $T\mathbb{H}^2$ and $\alpha =\alpha (x_1,x_2,y_1,y_2)$ is a positive function and $\rho$ is an arbitrary constant. $$ y_1\frac{\partial \alpha}{\partial y_2}-y_2 \frac{\partial \alpha}{\partial y_1}=0,$$ $$\frac{\partial \alpha}{\partial x_1}=0,$$ $$\frac{\partial \alpha}{\partial x_2}+\frac{1}{x_2}(y_1\frac{\partial \alpha}{\partial y_1}+y_2\frac{\partial \alpha}{\partial y_2})=0,$$ $$ \frac{x_2^2}{2\alpha}\frac{\partial \alpha}{\partial y_1}\frac{\partial \alpha}{\partial y_2}+x_2^2 \frac{\partial ^2\alpha}{\partial y_2 \partial y_1} + \frac{1}{2\alpha ^3}\frac{y_1y_2}{x_2^2}=0 ,$$ $$ \rho =-\frac{1}{\alpha}-\frac{x_2^2}{2}(\frac{\partial ^2\alpha}{\partial y_2^2 } +\frac{\partial ^2\alpha}{\partial y_1^2})-\frac{1}{2\alpha ^3}\frac{y_1^2+y_2^2}{x_2^2},$$ $$ \rho=-\frac{x_2^2}{2}(\frac{\partial ^2\alpha}{\partial y_2^2}-\frac{\partial ^2\alpha}{\partial y_1^2})-\frac{x_2^2}{2 \alpha}(\frac{\partial \alpha}{\partial y_2})^2+\frac{1}{2\alpha ^3}\frac{y_1^2}{x_2^2} ,$$ $$ \rho=-\frac{x_2^2}{2}(\frac{\partial ^2\alpha}{\partial y_1^2}-\frac{\partial ^2\alpha}{\partial y_2^2})-\frac{x_2^2}{2 \alpha}(\frac{\partial \alpha}{\partial y_1})^2+\frac{1}{2\alpha ^3}\frac{y_2^2}{x_2^2} .$$

Where $x_2\neq 0$ and we need to find $\alpha$ and we need $\alpha$ to be positive even locally.