Before revision:  We focus on leximin optimal fairness and show that finding such a matching is NP-Hard, even under isometric utilities. Next, we narrow our focus onto ranked isometric utilities and provide a characterisation for the space of stable matchings.
After revision:  We focus on leximin optimal fairness and seek leximin optimal many-to-one stable matchings. In the special case of ranked utilities where all agents on each side have the same rankings over the agents on the other side (but not necessarily the same utilities), we provide a characterisation for the space of stable matchings.