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The general solution of your equations in a simply connected domain on which $r_2\not=0$ and $r_1\not=\pm1$ is $$\beta = \frac12 + \frac1{{(r_1}^2{-}1)}\, \left(\frac{\partial a}{\partial\theta_1}+b(\theta_1,r_2)\right) \quad\text{and}\quad \gamma= \frac12 + \frac1{{r_2}^2}\, \left(\frac{\partial a}{\partial\theta_2}+c(\theta_2,r_1)\right),$$ where $a = a(\... 3 I would like to know why you are interested in the specific$d$-dimensional measures given by the Hausdorff ones. For$d=n$such a choice is understandable since the Lebesgue measure in$\mathbb{R}^{n}$is proportional to$\mathcal{H}^{n}$. But (to the best of my knowledge) there is no nice description in the case of$d\$-dimensional Hausdorff measures. A ...