TRENTON -- A New Jersey couple who gave their children Nazi-inspired names should not regain custody of them, a state appeals court ruled on Thursday.

The state removed the three small children of the couple, Heath and Deborah Campbell, from their home in January 2009.

A month earlier, the family drew attention when a supermarket near the family’s home in Holland Township, in west-central New Jersey, refused to decorate a birthday cake for their son, Adolf Hitler Campbell, now 4.

He and his sisters, JoyceLynn Aryan Nation Campbell and Honszlynn Hinler Jeannie Campbell, who are younger, have been in foster care.

A family court determined that there was insufficient evidence that the parents had abused or neglected the children, but a three-judge appeals panel, in reviewing that decision, determined that there was enough evidence not to return the children to their parents’ custody.

The panel sent the case back to family court for further monitoring.

A gag order remains in place and the parties refused to discuss the decision.

Last year, Heath Campbell accused the state of removing the children because of their names and said government officials were relying on unproven accusations made by a neighbor and by an ex-wife who charged him with abusing her years ago.

In the ruling on Thursday, which did not mention the names or the birthday cake, the court found that there were myriad reasons that proved the need for continued protection services for the children.

According to court records, both parents are unemployed.

The court found that both parents were themselves victims of childhood abuse and said that they had not “received adequate treatment for their serious psychological conditions.”

Mr. Campbell, 37, cannot read, and Ms. Campbell, who is in her mid-20s, left school before finishing the 10th grade, according to court records.