Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms

We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic space forms $\mathbb{H}{P}^m$ and $\mathbb{H}{H}^m$ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius $r\in (0,\pi \mathrm{/}2)$ except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a canonically embedded $\mathbb{C}{P}^m\subset \mathbb{H}{P}^m$. On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in $\mathbb{H}{H}^m$ is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane $\mathbb{H}{H}^{m-1}\mathrm{.}$ Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere $H_3$ in $\mathbb{H}{H}^m$ is not of finite type but satisfies ${\mathrm{\Delta }}^2\tilde{x}=$ const.