This paper presents an approach for elimination of anomalies in quantum theory of open Nambu-Goto relativistic strings, formulated in Minkowski space-time of dimension d=4. Clue idea is a choice of the light cone gauge, whose gauge axis is related with the world sheet in Lorentz-invariant way. In this approach Lorentz transformations act both on the worldsheet and the gauge axis, and they are not followed by reparametrizations. As a result, the theory becomes free of anomalies in Lorentz group and in the group of internal symmetries of the system. It is shown that gauge conditions of Abelian type lead to algebraically simple Hamiltonian mechanics, convenient for canonical quantization. The constructed quantum theory possesses spin-mass spectrum with Regge-like behaviour.

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