Thanks for your question.
The starting point if you are both jointly named is that both of your have a right of occupation which neither can deny the other unless there is a Court order to contrary pursuant to criminal or legal proceedings.
If you are jointly named then in the absence of a declaration of a trust declaring the precise shares there is a presumption that you each own a 50% share of the equity in the property.
If you move out and continue to pay your share of the mortgage repayments then the amount of yoru interest in the property will not chance and will remain presumed at 50%.The presumption can be rebutted but the burden will be on the person claiming more of the equity to show why it should differ (eg. capital improvements paid which increase the value of the property, capital repaymenst of the mortgage etc).
If you move out you would still be able to apply for an order for sale to realise your equity. The situation is that while the Courts are reluctant for persons to remain on registered titles to the properties complicated to an extent in that the primary consideration of the Court will be that all the children's accomodation needs are met. If it is the case that the party to have residence of the children shall be able to adequately re-house the children then an order for sale would probably be made. So you situation is dependent, to an extent, on your partner remaining willing to sell or being able to rehouse herself and the children adequately.
Litigating in this regard can become expensive and where possible it is generally best to settle unless the equity is very large or where one party is being wholly unreasonable in their demands.
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