As a supervisor that doesn't get any support from her manager, how should I stay focus and inspire my team?

Always keep the respectful attitude intact despite your Supewrvisor being non-suppotive and never break the chain of command. Ensure your team is not aware of the your Supevisor does not support or else it will demorlize the teamwork.  

should maintain supportive nvironment and focus on work and keep following the SOP

widing up my own knowledge and experience, holding responsibilities and make them feel safe by leading them not just controlling them

First of all we should bring a friendly atmosphere among the members.Make sure that will not effect the task.As a supervisor learn the member deeply and understand their feelings.Help the members to solve their  problems as a good friend and leader.If we are not getting  any management support, share among the members ,they will also understand the problems and work according to that.Members will help even if there is no management support to achieve the task.

Sometimes hands off style from managers is helpful to supervisors because it will develop the abilities in problem solving and skills to negotiate with other colleagues. Just do your job like leading your teams, plan & organize work with your teams and help others by trainings & practices. Through these, other people will admire you and even follow your track.