Abstract: We construct new conformal minimal 2-spheres of constant curvature linearly full in the quaternion projective space $\mathbb{H}{{\mathbf{P}}^{n}}$ by the twistor map $\pi :\mathbb{C}{{\text{P}}^{2n+1}}\to \mathbb{H}{{\text{P}}^{n}}$ in a systematic way. That is, we construct 2-spheres linearly full in $\mathbb{C}{{\text{P}}^{2n+1}}$ or $\mathbb{C}{{\text{P}}^{2n}}$ satisfying horizontal condition explicitly. (In the latter case, we consider the natural embedding of the map). We prove all 2-spheres we constructed are non-homogeneous.

Key words: quaternionic projective space, harmonic sequence, homogeneous 2-sphere, constantly curved minimal 2-sphere