Some people get confused about job-sharing, which has no formal existence in employment law. A job-share is nothing more than the employment of two part-timers, whose combined contracts might otherwise be undertaken by one full-timer.
They are separate employees and their terms of employment may be varied individually. The resignation of one does not deprive the other of a job.
You are, therefore, free to deploy these two part-timers in whatever way is convenient, subject to the terms of their contracts. Those contracts can also be revised by mutual agreement.
A parent has written to say that she wishes her son's name to be changed on the school register from that of her ex-husband to her own. Should we comply and how does this affect official documents, such as examination entries?
The law relating to names is, in some ways, conflicting. On the one hand, it is accepted that a person is entitled to be known by whatever name they choose. On the other, for official purposes one bears the name that appears on one's birth certificate and this can only be changed by obtaining the legal document known as a Deed Poll.
You need to approach the present case with caution, because both you and the boy concerned may be victims of exploitation by one party to an ex-matrimonial dispute.
It is perfectly permissible to change the name by which the boy is known in the school but it might be useful to talk to the mother about the question of public examination certificates.
If we are talking here about a secondary school pupil, it might not be a bad idea to check on what the boy himself wishes. It is his name, after all.