This note further carries on the study of the eigenfunction problem: Find f(t) = p(t)eiθ(t) such that Hf= - if, p(t) ≥ 0 and θ(t) ≥≥ 0, a.e. where H is Hubert transform. Functions satisfying the above conditions are called mono-components, that have been sought in time-frequency analysis. A systematic study for the particular case ρ = 1 with demonstrative results in relation to Möbius transform and Blaschke products has been pursued by a number of authors. In this note, as a key step, we characterize a fundamental class of solutions of the eigenfunction problem for the general case ρ≥ 0. The class of solutions is identical to a special class of starlike functions of one complex variable, called circular H-atoms. They are building blocks of circular mono-components. We first study the unit circle context, and then derive the counterpart results on the line. The parallel case of dual mono-components is also studied.