obtaining a "residence" permit for an over-60 teacher

Nothing in the English version of the official Chinese regulations for obtaining a "Z" visa mention an age restriction. However, some Foreign Expert Bureau Offices adhere to an age 60 limit. Consequently, most schools will not employ a teacher over 60 for fear of being unable to obtain the "Z" visa for the teacher. The annual "residence" permit appears to be a way around this restriction. The question is, how does one apply for a "residence" permit without an employer? Many schools have said they would hire qualified experienced over-60 teachers IF the residency issue were resolved. Again, how does one obtain a "residence" permit?