  Periodic solutions of the Manakov system are studied with the aim to
establish links between recent results of Wright (Physica D 2013 {\bf 264} 1)
and Kamchatnov (EPL 2013 \textbf{103} 60003), where the one-phase solutions
have been obtained by different methods and with different parameterizations.
Two types of waves, which in the context of dynamics of two-component
Bose-Einstein condensates can be called {\it density} and {\it polarization}
waves, are determined as particular solutions of the Manakov system. Whitham
equations describing modulations of these two types of waves are obtained.
