Stegall compact spaces which are not fragmentable

Using modifications of the well-known construction of “double-arrow” space we give consistent examples of nonfragmentable compact Hausdorff spaces which belong to Stegallʹs class S. Namely the following is proved. ℵ1 is less than the least inaccessible cardinal in L and MA & ¬CH hold then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a Baire space into K is singlevalued at points of a residual set. V=L then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a completely regular Baire space into K is singlevalued at points of a residual set.