Two Casey Anthony convictions thrown out

(CNN) -- One of the most infamous Americans in recent years took two big steps Friday toward clearing her name, and she's vowed to "keep fighting."

Florida's 5th District Court of Appeal threw out two of Casey Anthony's four convictions of "providing false information to a law enforcement officer during a missing person investigation," agreeing with her argument that the multiple convictions violated the ban on double jeopardy.

But the appeals judges upheld the other two convictions. According to the court filing, they rejected Anthony's claim that the trial court should have granted her motion to suppress statements made to law enforcement officers before they told her her Miranda rights. And they rejected her argument that the state statute she was convicted of violating is unconstitutionally vague.

Anthony's lawyer, Cheney Mason, said when he called his client to share the ruling, she said, "We keep fighting."

Anthony could appeal the remaining two convictions to the Florida Supreme Court next.

The four charges arose from statements Anthony made on July 16, 2008, to Orange County Detective Yuri Melich, who was investigating the disappearance of Anthony's 2-year-old daughter, Caylee.

Anthony gained notoriety after her mother, Cindy Anthony, called police to arrest her daughter the day before, when she thought Casey had stolen the family car and some money.

During the call, Cindy Anthony told police she was concerned that her granddaughter might be missing, and said, "I found my daughter's car today, and it smelled like there's been a dead body in the damn car."

When questioned, Casey Anthony admitted to police that she hadn't seen Caylee for more than 30 days, and on July 16, she was arrested on suspicion of child neglect, filing false official statements and obstructing a criminal investigation. At that time, she made the statements to Melich, which led to her convictions.

In December 2008, a utility worker found remains in a wooded area near the Anthony home, and about a week later, authorities announced they were Caylee's.

After nearly three years of legal maneuvers, Anthony's capital murder trial began on May 24, 2011.

Prosecutors alleged that she killed Caylee by using chloroform and covering her nose and mouth with duct tape, and that she put her body in the trunk of her car before dumping it in the woods.

Defense attorney Jose Baez argued that Caylee drowned in the Anthony family pool on June 16, 2008, and that Casey Anthony and her father, George, covered up the death.

On July 5, 2011, a jury found Anthony not guilty of first-degree murder, aggravated child abuse and aggravated manslaughter of a child, while convicting her on the four "false information" counts.

Anthony was sentenced to four years in jail, to be served consecutively, including time served. She was released 10 days after her sentencing, on July 17. Even though she exited shortly after midnight, about 1,000 people jeered her as police guards and Baez escorted her to a waiting SUV and drove away quickly.

Anthony still faces a defamation lawsuit from Zenaida Fernandez-Gonzalez, of nearby Kissimmee, Florida.

During Anthony's disputed July 16, 2008, statements to Melich, she said the last time she had seen her daughter was when she dropped Caylee off at Gonzalez's apartment.

Gonzalez filed suit in September 2008, claiming that Anthony ruined her reputation.

Last April, a judge ruled that the suit needed to go to trial by jury and denied Gonzalez's request for a summary judgment.