Two approaches to anomaly-free quantization of general covariant systems on an example of a two-dimensional string

In this paper we discuss two approaches to anomaly-free quantization of a two-dimensional string. The first approach is based on the canonical Dirac prescription of quantization of degenerated systems. At the second approach we "weaken" the Dirac quantization conditions requiring the solving of first class constraints only in the sense of mean values. At both approaches there are no states with the indefinite metrics.