Sanjeev Chada, 43, has been accused of killing 10-year-old Eoghan and five-year-old Ruairi and was remanded in custody to Castlerea Prison, County Roscommon.
He appeared at Harristown District Court this morning where the need for a psychiatric assessment was raised by his solicitor James Hanley, who also asked that Chada be placed on suicide watch in prison.
The judge has directed appropriate medical treatment for Chada, who will appear at Harristown Court again on August 16.
Their bodies were discovered the next day in the boot of Chada's crashed car, four miles from Westport, Mayo.
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Irish detectives have sealed off a second crime scene close to Ballintubber Abbey in the county where it is believed the boys died of asphyxiation, which can be caused by strangling or suffocation.
The father, who has been described by his neighbours as a "prefect devoted dad", is believed to have strangled the kids.
The boys' mother, Kathleen Chada, a nurse, has been comforted by relatives, including her parents Willie and Patsy Murphy, since the alarm was raised in the early hours of Monday morning when they did not come home from a bowling trip.
Gardai, as the Irish police is known, issued the first-ever amber alert for missing children - the Child Rescue Ireland Alert - when grave concerns over their safety was raised.
It is understood Chada originally hails from the north-east of England with family connections in the Middlesbrough and Newcastle areas, but his mother has spent recent years living in London.