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The U.S. Supreme Court this week faces the first religious objection to reach the justices since the decision declaring a constitutional right to same-sex marriage.

A county clerk in Kentucky is asking for relief from a lower court order directing her to issue marriage licenses to gay and lesbian couples. Her lawyers filed an emergency application directed to Justice Elena Kagan, who handles such requests from that part of the country.

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Related: Kentucky Clerk Asks Supreme Court to Intervene in Gay Marriage Case

Kim Davis, the clerk in Rowan County, Kentucky, "holds an undisputed sincerely held religious belief that marriage is a union between a man and a woman, only," her lawyers say in their application.

The requirement to issue same-sex marriage licenses "demands that she either fall in line (her conscience be damned) or leave office (her livelihood and job for three-decades in the clerk’s office be damned)."

Immediately after the Supreme Court's June 26th gay marriage ruling, Kentucky's governor, Steve Beshear, ordered all the state's county clerks to comply with the decision and issue marriage licenses to same-sex couples.

When Davis refused, citing her religious objection, couples seeking licenses sued, and a federal judge ordered her to comply. Last week a three-judge panel of the Sixth Court of Appeals agreed.

"It cannot be defensibly argued that the holder of the Rowan County Clerk’s office, apart from who personally occupies that office, may decline to act in conformity with the United States Constitution as interpreted by a dispositive holding of the United States Supreme Court," the appeals court said.

Justice Kagan could act on the request herself or refer the application to the full court.