  The two ways of constrained systems quantization are considered from the
point of view of their self-consistency at the quantum level. With a
transparent example of a particle in the external electromagnetic field we
demonstrate that the procedure of gauge fixing turns out rather dangerous and
may lead to a quantum anomaly in the operator algebra. We discuss additional
classical symmetries as an essential element for tracing out this anomaly. The
two cases of a spinning and a spinless particles in the external
electromagnetic field are discussed to illustrate the situation.
