I am a divorced mum, an academic, in Israel, I have just been offered a fantastic 3 year research posting overseas but my 'ex' is against our child (who lives with me) leaving Israel. What can I do?

Apply to the family court for permission to relocate overseas with your minor child for a specified period of time. Where parents cannot agree about where a child should live, even temporarily, then the family court hears the parties' competing claims, and can rule on the dispute. It has discretion to overrule the non-custodial opposition's opposition to relocation overseas.
Abduction From Israel