Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps II

We further study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces, extending the results found in Gonçalves and Kelly (2010) [6], where a number of results are obtained for the case when the target surface is the Klein bottle. Here the target will be a closed surface of Euler characteristic <0, and also a new result for the Klein bottle. For this problem we study certain equations on the pure braid group on two strings of the target surface. With the target surfaces stated above we give a classification according to when the answer of the two problems is either the same or different.