Neumann property in the extended modular group and maximal nonparabolic subgroups of the modular group

It is know that any Neumann subgroup of the modular group is maximal non-parabolic. The question arises as to whether these are the only maximal non-parabolic subgroups. The wild class of maximal, non-parabolic, not Neumann subgroups of the modular group was constructed by Brenner and Lyndon. The new construction of such a class is presented. Those groups are obtained as subgroups of elements of positive determinant of any Neumann subgroup of the extended modular group (the notion which we introduce in the paper) and in this sens they are as “close” as possible to Neumann subgroups of the modular group.