  In this paper we consider ultra-parallel complex hyperbolic triangle groups
of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic
plane, generated by complex reflections in three ultra-parallel complex
geodesics two of which intersect on the boundary. We prove some discreteness
and non-discreteness results for these groups and discuss the connection
between the discreteness results and ellipticity of certain group elements.
