Half-homogeneous chainable continua with end points

A point of a chainable continuum is called an end point if for every positive epsilon there is an epsilon-chain such that only the first link contains the point. We prove that up to homeomorphism there are only two half-homogeneous chainable continua with two end points. One of them is an arc and the second one is the quotient of an arc of pseudo-arcs, where the two maximal continua containing only end points are pushed to points. This answers a question of the second and third authors.Moreover we prove that the two above mentioned continua are the only half-homogeneous chainable continua with a nonempty finite set of end points.