The 39-year-old father was arrested and charged with manslaughter, negligent murder and endangering the well-being of a child.
"The family (Rodriguez) is being torn apart," Jackson said. "His state of mind is very fragile, based on what happened, it's just a terrible scenario."
Rodriguez is being punished for careless murder and manslaughter with up to four years' imprisonment and manslaughter of up to 15 years imprisonment. He was charged on Saturday with a $ 100,000 bail of $ 50,000, a source told the Central Booking Clerk's Office in Bronx.
Rodriguez "never intended to let this happen," says the lawyer.
The twins Mariza and Phoenix Rodriguez were found to be unresponsive in the backseat of a Honda limousine in the Bronx, the NYPD said. They were declared dead at the scene.
The New York coroner will investigate the cause of death of the twins, the police said.
While the twins were in the car, their father was at work in a nearby Virginia hospital.
Jackson said his client had "never intended to let this happen – it's such a pity that this is happening all over the country, too often we report on it "[Luckilyhehasatremendoussupportnetworkworkingwithhim-withhisfamilyhiswifehisfriends"Jacksonsaid"Hehas5kids-now3-andeveryoneisupset"
The prosecution does not think the incident was intentional, Jackson said, but they say he was reckless and negligent.
"The claim is that he was extremely negligent," Jackson said.
The legal plan for the moment is to appeal to DA to resolve the situation regarding justice and the grief and pain of the family, Jackson said.
"What are you going to do with the guy, throw him in jail and take him away from his family and the rest of the children, what is the appropriate solution for this type of case?" Jackson said.
A judge asked to put Rodriguez on suicide watch, Jackson said during a Saturday night press conference. Rodriguez & # 39; s next court date is August 1, when a large jury decides whether to file an indictment or not.
The "Father of All Fathers, Father of All Fathers"
Jackson said that before he was called to represent Rodriguez, the two did not know each other.
However, they know each other, common friends, who call Rodriguez "the father of all fathers, the father of all fathers," Jackson said.
Rodriguez is a social worker at the hospital, his friend Temple Barros (41) told CNN.
"He's always been a great dad, whatever he needs, he'll go out and get it," Barros said.
He said the family has a "very smart" older daughter. The mother "does not believe what happened," he said. "Her parenting is amazing, I'm at a loss."
Hospital spokesman Jim Connell confirmed that Rodriguez is an employee and stated:
"The entire hospital community is saddened by the news of this tragic event The situation is currently under the jurisdiction of local authorities We express our sincere and heartfelt condolences to the family. "
CNN's Elizabeth Joseph reported from New York, and Amir Vera reported and wrote from Atlanta; CNN's Josh Girsky, Tatyana Bellamy-Walker, Laura Ly and Sheena Jones contributed to the report.