When Argentina plunged into a major economic crisis almost a decade ago, thousands of its citizens fled abroad.

Many, like the Flores family, resettled in South Florida but were not able to obtain proper residency documents.

Now, U.S. immigration officials are gradually discovering them.

On Dec. 14, Immigration and Customs Enforcement agents arrived at the Flores family’s North Miami home and detained Rubén Flores, 46, and his wife, Claudia Quevedo, 44. When their daughter Rocío, 22, arrived to see what was happening, she, too, was detained.

The family arrived in South Florida in 2001, when visitors from Argentina could enter the United States without a visa. Even so, that did not mean they were free to stay indefinitely.

Eventually, they were fitted with electronic ankle bracelets and released from detention. Immigration authorities told them they must return to Argentina by this Friday, Rocío Flores said. However, an ICE spokeswoman, Danielle Bennett, told El Nuevo Herald that her agency has no plans to deport Flores or his family.

“If the deportation is carried out, this is going to destroy the family and ruin what we built during the last 10 years,” said Rocío, who came to South Florida with her parents, a brother, Edgard, and a sister, Danina. The children attended high school in South Florida. Flores later attended Miami Dade College.

Another younger brother, Dilhan, 9, was born here.

The family worked at the Mario the Baker pizza restaurant in Bay Harbor Islands for three years. In 2010, they bought it.

Now, Flores worries that if her parents are forced to return to Argentina, the business will be shuttered and its employees will lose their jobs.

ICE’s posture toward the Flores family seems to confirm a new Obama administration policy of deporting foreign nationals with criminal convictions rather than undocumented immigrants with no police records.

But, federal officials say, this doesn’t mean detention of undocumented immigrants will cease. Detentions will continue, though fewer of those without criminal convictions will be deported, they say.

Even when detainees are released, many will not be able to resolve their immigration status because they cannot apply for permanent residence or citizenship.

In the case of Rocío Flores and her sister, both have married U.S. citizens and can apply to become legal residents . But in other cases, the most that undocumented immigrants who have been released from detention can do is apply for a work permit.

South Florida’s Argentinian community is the largest in the United States, according to Argentine diplomats in Miami.

In July, the case of Miyen Spinelli drew attention after the Argentine consul in Miami asked ICE not to deport him. Spinelli arrived in 2001 with his family when he was 13.

The federal government decided Spinelli could stay.

The case of undocumented Argentines is particularly complex because many arrived during a time when visas were not necessary to enter the country.

A foreign visitor admitted without a visa has fewer rights if detained for being in the country illegally than do those with visas. For example, foreigners who have entered the country legally without a visa, as Argentines did in 2001, have no right to have their case heard in immigration court.