In the rules for writing a detective story what does it say about a love interest for the Detective?

There are no "Rules for Writing a Detective Story!"

Detective stories are just stories. The idea of a detective or detecting consultant was invented by Arthur Conan Doyle with his Sherlock Holmes stories,

No, it was invented by Edgar Allan Poe in his three C. Auguste Dupin stories: "The Murders in the Rue Morgue," "The Mystery of Marie Roget," and "The Purloined Letter."

and the idea became popular with the readers.

Nowadays, you can find many different types of mystery and detective stories. Read as many different ones as you can, and you will see different ways that the authors have handled the love interest for their characters.