A mother who fled with her two sons drugged them with sedatives and made allegations of sexual abuse against their father, a judge has said.
Samantha Baldwin, 40, and her sons vanished on 27 March at the conclusion of a family court hearing.
Judge Jeremy Lea said that although she had genuinely believed there had been sexual abuse, her allegations were false.
Ms Baldwin has been arrested on suspicion of abduction and bailed.
The mother and boys were found on 6 April near Ollerton in Nottinghamshire.
The judge said Ms Baldwin took the boys in a move that was "plainly pre-planned and carefully executed".
He said there had been "ill-informed speculation" about the case and he was now setting out the facts.
A hearing that began in February heard 12 days of evidence.
Judge Lea said: "In my judgement I found that the father had not perpetrated sexual, physical or emotional abuse against the boys as alleged by the mother, or at all, that the allegations made by the mother were false.
"I also found that the mother genuinely believed that he had done so, but that her belief was irrational and that the evidence of abuse was unreliable.
"I also made a finding that in order to try to prove her case against the father, the mother had caused the boys to ingest substances that would give rise to a positive testing for benzodiazepine products and zolpidem".
Benzodiazepine and zolpidem are used as sedatives and anti-anxiety treatments.
The judge also noted the effect of the case on "the emotional and psychological well-being of these boys may well be catastrophic".
The boys are currently in local authority care.
In a statement, the boys' father thanked the court for its judgement and said: "My reputation and that of my family and friends has been wrongly and unfairly tarnished with the most distressing and false allegations.
"The effect of this on [the boys] however, has been much greater and we now have to focus on piecing our lives back together."