1/n-homogeneous, circle-like, indecomposable continuum, curve of pseudoarcs

A continuum X is called solenoidal if it is circle-like and nonplanar. X is 1/n-homogeneous if the action of its homeomorphism group on X has exactly n orbits; i.e. there are exactly n types of points in X. Recently Jim\'enez-Her\'andez, Minc and Pellicer-Covarrubias [Topology and its Applications, 160 (2013) 930--936] constructed a family of 1/n-homogeneous solenoidal continua, for every n>2. Modifying the spaces obtained by them, as well as an earlier construction of the author for n=2, for every n>2 we construct two different uncountable families of arcless 1/n-homogeneous solenoidal continua. We also show that there is an uncountable family of countably nonhomogeneous solenoidal continua. With respect to the degree of homogeneity, in the realm of solenoidal continua containing pseudoarcs, our examples complete the gap between homogeneous solenoids of pseudoarcs and uncountably nonhomogeneous pseudosolenoids. A number of questions related to the study is raised.