This question is purely theoretical, however, the situation would be something along the lines of prints on the murder weapon or the victims blood on the suspects clothes, however the suspect would have no specific reason to commit the crime.
Theoretically if there is physical evidence that a murder (homicide) was committed and that a particular person committed it, I would expect that the person would be arrested and charged. Whether the charge would be murder or manslaughter depends on what the prosecutor thinks that they could prove at trial. As to lack of a specific reason, I can't imagine that there would be no possible motive that a prosecutor could argue to the jury.
Motive is a factor but is not a definitive defense. You would be surprised at the creativity of DA's in coming up with a "motive". The jury can consider that a defendant claims to have no motive, but it would not require a dismissal.
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