The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum

A set is called a selector of an equivalence relation defined on all the real numbers if it intersects each equivalence class of this relation in a singleton set. The following proposition is called the selector principle: each analytic equivalence relation on the set of all real numbers has an $A_2$-selector. It is proved that the selector principle is not equivalent to the existence of an $A_2$ well ordering of the continuum. This answers a question posed by Burgess. Equivalence is understood in the sense of equivalence in the standard Zermelo–Fraenkel set theory with the axiom of choice.
Bibliography: 8 titles.