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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.

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Your problem can be simplified considerably if you use the fact that the first equation and the second equation imply that $\alpha$ has the form $\alpha = f(x_2,r)$ where $r = {y_1}^2+{y_2}^2$. Then the third equation implies that, in fact, $\alpha$ must have the form $$\alpha(x_1,x_2,y_1,y_2) = g\left(\frac{r}{{x_2}^2}\right) = g\left(\frac{{y_1}^2+{y_2}^2}{{x_2}^2}\right)$$ for some function $g$ of a single variable.

Now, the remaining equations (which are second order) will give you a set of ODE for $g$, and you should easily be able to tell whether they have solutions.

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