Detectives said they were not looking for anyone after a surgeon and his son were found dead at their home.
Adel Makar, 60, and his son Emet, 26, were found at a house in Westcliff-on-Sea, Essex, on 22 March.
Post-mortem tests showed Mr Makar had been stabbed and his son had died from strangulation, police said.
The spokesman said officers were not making any witness appeals and were not actively seeking any third party in connection with the incident.
One theory detectives are pursuing is that Emet Makar killed his father before committing suicide.
"We were alerted by a relative and discovered the two bodies in the house," said the spokesman.
"We have not made any appeals for witnesses but we are continuing to investigate the circumstances surrounding the deaths."