The Santa Fe County Sheriff says he's not ruling anything out in the deaths of iconic actor Gene Hackman, his wife, Betsy Arakawa, and their dog -- and his deputies even got a search warrant in case the investigation turns criminal, TMZ has learned.

Sheriff Adan Mendoza tells TMZ ... his deputies responded Wednesday to the compound of Hackman and Arakawa after receiving a call ... and they initially entered the house only to find the bodies of the couple and one of their 3 dogs -- 2 others survived.

Once they found the corpses, Mendoza says, the deputies backed out of the house and applied for a warrant to conduct a more thorough search.

Mendoza says it's customary during death investigations to seek a warrant on the assumption they may find evidence of a crime, which shows they're following standard procedure.

Mendoza says the gas company and fire department also responded to make sure there were no toxic fumes that would endanger the deputies during their search.

The sheriff says deputies determined the bodies had been there for at least a day ... and he did not rule anything out, although no visible trauma was discovered. Mendoza says they could have been victims of a double homicide, suicide, accidental death, or natural causes.

Mendoza says it's unclear what will happen with the two surviving dogs. We'll update you when we get more.