Is there a way (can I hope) to express the solutions of the following coupled transcendental equations in a closed form (in terms of Lambert, hypergeometric or other special functions) : \begin{eqnarray} \phi_{1}=\operatorname{Im}[\Psi[A_{2}+i B_{2}-i C_{2} \phi_{2}]]\\ \phi_{2}=\operatorname{Im}[\Psi[A_{1}+i B_{1}-i C_{1} \phi_{1}]] \end{eqnarray} All constants ($A$'s, $B$'s and $C$'s) are real and positive. And $\operatorname{Im}[\Psi[z]]$ stands for imaginary part of digamma function of $z$. As is clear from the equations I am looking for real solutions $(\phi_{1},\phi_{2})$ only.
Thanks in advance.