This question is related to the existence of Einstein metrics on tangent bundles where the metric is induced by the [isotropic almost complex structures](http://link.springer.com/article/10.1007/BF02568316) 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.



Solutions of this system of PDE's
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.


