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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 $$ \beta = \frac12 + \frac1{{(r_1}^2{-}1)}\,\frac{\partial a}{\partial\theta_1} \qquad\text{and}\qquad \gamma= \frac12 + \frac1{{r_2}^2}\,\frac{\partial a}{\partial\theta_2}, $$ where $a = a(\theta_1,\theta_2)$ is a function of $\theta_1$ and $\theta_2$ only. I do not see how you can choose $a$ so that your 'boundary conditions' are satisfied.