How do i explain to the immigration that I’m not legally married to the mother of my kids but she bears my surname?

Tell them, that there was never any marriage simple 

Explaining your marital status to immigration authorities can be straightforward if you provide clear and honest information. Here's how you might explain your situation:

1. Be Honest: Start by being honest about your marital status. Clearly state that while you are not legally married to the mother of your children, she bears your surname.

2. Provide Context: Explain the nature of your relationship with the mother of your children. Describe how you have a familial relationship and share responsibilities as parents, but you have not formalized your relationship through marriage.

3. Explain Cultural or Personal Reasons: If there are cultural or personal reasons why you have not legally married but still maintain a family unit, you can mention these to provide context to your situation. This might include cultural norms or personal beliefs about marriage.

4. Offer Supporting Documentation: If applicable, provide any supporting documentation that confirms your relationship, such as birth certificates of your children listing you as the father, shared bank accounts, joint leases or mortgages, or affidavits from friends or family who can attest to the nature of your relationship.

5. Clarify the Use of Surname: Explain that the use of your surname by the mother of your children is a personal or family choice, but it does not imply a legal marriage.

6. Answer Questions Truthfully: If immigration authorities have specific questions about your relationship or marital status, answer them truthfully and provide any additional information or documentation they request.

7. Seek Legal Advice if Necessary: If you're uncertain about how to navigate the immigration process or if you have concerns about your specific situation, consider seeking advice from an immigration lawyer in your country, who can provide guidance and support.

Overall, the key is to be transparent and forthcoming with information while providing context to help immigration authorities understand your family dynamics and marital status accurately.