In September 2010, the pair were taken from their home
in Snowmass, Colorado, and brought to Buenos Aires, Argentina, by their mother, Ana Alianelli. Alianelli is an Argentine native.
At the time, Victoria was 4 and Sophia was 2. The children were caught in the middle of an international parental abduction that resulted in a legal nightmare for their father.
"I think the girls right now are very confused," Burns told CNN in a recent interview. "In time, they're going to see that having a mother and father in their lives is the most important thing."
Burns and Alianelli were once happily married, but their divorce proceedings grew bitter. Alianelli requested to relocate the children to Buenos Aires, and her request was shot down. After a 13-month custody battle, a Colorado judge ruled Burns as the primary residential parent and said the children would remain in the United States.
Three weeks later, against the judge's orders, Alianelli took the girls to Buenos Aires.
Argentina is a member of the Hague Convention on the Civil Aspects of International Child Abduction, which is meant to help promptly return children who have been abducted by a parent. But Burns' battle has dragged on for years.
The case was passed from one Argentine court to another due to a litany of appeals made by Alianelli's lawyers.
The case eventually landed in the Supreme Court of Argentina, the country's highest court, which rejected Alianelli's final appeal last fall.
"I'm tired of battling legally," Burns said. "I'm exhausted from the years of emotional pain. I look forward to happier days now."
Burns believed those happy days were within arm's reach. He was told to travel to Argentina in late March for the judge to sign the final order of return.
Alianelli has denied previous requests to speak with CNN, but when Burns arrived to sign the final return order, Alianelli took to Argentine television. She accused Burns of being a drug addict and abusive, claims she has made in the past and claims that a U.S. court-appointed investigator found to be "fabricated."
Finally, on April 13, the order was issued. Burns went to the U.S. Embassy in Buenos Aires and picked up Victoria and Sophia, his daughters solely in his custody for the first time in more than 4½ years. Victoria is now 8 and Sophia is 6. Sophia has spent so much of her life in Buenos Aires that she is not fluent in English, making communication with her father a challenge.
Both parents agreed that it would be in the best interest of the children for Alianelli to join them on their return to the United States.
It seemed the ordeal was finally coming to an end, but at the airport in Buenos Aires on April 15, Burns encountered a scene he didn't anticipate. He and his children were swarmed by local Argentine media, yelled at by protesters, and friends of the girls chanted their wish for Victoria and Sophia to stay.
As family tensions boiled over, Alianelli's brother, Javier, attacked Burns, jumping on him and putting him in a chokehold. The scene grew chaotic; Burns lost his grip on Sophia in the fray. She was picked up by an airport employee, and Burns and his children were whisked away as both girls cried hysterically.
Fourteen hours later, CNN was at the airport in Aspen, Colorado, awaiting their arrival. But only Alianelli got off the plane.
"This is the thing, the story is not only one side," she said. "The girls are missing right now."
The girls were with Burns in Houston, Texas. Burns said that after he and his daughters arrived in Houston, their first layover stop, he sought medical treatment for his neck. Alianelli apparently wasn't aware.
Victoria and Sophia are now back in Snowmass, Colorado, living with Burns. In the first week since their return, Burns has taken the girls bowling, shopping, to a horse ranch, and of course, to the Colorado mountains to build snowmen. But the legal battle is not yet over.
With Alianelli back in the U.S., the Colorado judge who handed out the original custody decision will have to reconsider what type of access to the children Alianelli can be granted. For now, she is allowed supervised visitation only.