Following an exercise in Uniformly Most Powerful tests and the correspoding Neyman–Pearson lemma, it is possible to show that a certain test exists, where the critical values $c_1$ and $c_2$ can be solved under a certain $\alpha$ using the following system of equations:
$$\left\{ \begin{array}{rl} \mathbb{P}(c_1 < \chi^2_{2\lambda} < c_2) = 1 - \alpha \\ \mathbb{P}(c_1 < \chi^2_{2\lambda+2} < c_2) = 1 - \alpha \end{array} \right.$$
Here $\lambda > 0$ is known.
However, I'm not sure how to proceed: in one way, this corresponds to
$$\left\{ \begin{array}{rl} F(c_2) - F(c_1) = 1 - \alpha \\ G(c_2) - G(c_1) = 1 - \alpha \end{array} \right.$$ where $F$ and $G$ are the CDF's of $\chi^{2}_{2\lambda}$ and $\chi^{2}_{2\lambda+2}$, but I can't see a way to extract the values using only inverses. On the other hand, this then seems like an integral equation
$$\left\{ \begin{array}{rl} \int_{c_1}^{c_2}dF = 1 - \alpha \\ \int_{c_1}^{c_2}dG = 1 - \alpha \end{array} \right.$$ but not having much experience with integration, I'm not sure how are such equations usually solved.
Any tips or suggestions would be appreciated!