Direct speech sentences without leading "He said,", "She said,", etc
He declined the invitation to come to the party, "My cat is ill, so I have to be at home tonight."
This is not strictly correct but, if you are an author writing a novel then you are allowed a certain amount of artistic licence. If you consistently do so, your readers will get used to it. They may like it or loathe it but they will get used to it (or stop reading!).
Other possibilities than repeating "he"
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He declined the invitation to come to the party by saying,"My cat is ill, so I have to be at home tonight."
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He declined the invitation to come to the party, saying, "My cat is ill, so I have to be at home tonight."