Thanks for your question.
I assume that when you state " multiple visa" you are referring to a Multiple Entry Visitor's visa permitting him entry to the UK on multiple occasions throughout the course of the visa.
This being the case while it is physically possible for you to marry at a Registry office in the UK whilst here on that visa I would not generally advise you to do so. This is because the visa is for visitor purposes only. There is a special visitor's visa called a Marriage Visitor's Visa for visitor's who wish to come to the UK in order to marry:-
See Visitor's for the purpose of marriage at the bottom of the above link
If it is intended that he shall remain in the UK after marriage to settle permanently with you then he could have to apply for settlement before he came, which is different.
If it is intended that he marries here then returns to Russia for a period of time before returning to the UK to settle permanently with you later then at that later time he would have to apply for settlement. If you require guidance on the eligibility criteria and basic supporting documentation/waiting times then please submit a further question to the bard again requesting me via my profile and I shall assist.
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What I mean is that you could go to a registrar when he is on his multi-entry visa and get married provided you can prove your identities and put the appropriate notice of your marriage up at the registry office.
So it is possible. However, the fallout with the UKBA should be considered. Because the correct form of visa to marry in the UK on a visit is the marriage visit visa above the UKBA will not be happy that the marriage has occured in breach of the multi-entry visa and therefore they may take an awkware/adverserial approach to his settlement application when does make it in the future. There motivation to reject will be increased as a result and it could cause problems in the long term.
Were you planning on marrying here and then moving to Russia to settle then it would not be a problem but if the plan is to ultimately settle in the UK then I would advise that he applies for the marriage visitor's visa.
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