A Factorization Problem on a Smooth Two-Dimensional Surface

Given a continuous complex-valued function $a$ and nonnegative functions $\rho_1$ and $\rho_2$ on a two-dimensional smooth connected closed surface such that $|a|=\rho_1\rho_2$ and the functions $\rho_1$ and $\rho_2$ have no common zeros, it is required to find complex-valued continuous functions $a_1$ and $a_2$ satisfying the conditions $a_1a_2=a$ and $|a_j|=\rho_j$. Necessary and sufficient solvability conditions for this problem are given.