This paper is the first attempt to study the question of completeness of systems of axioms in the algebra of algorithms which, in addition to closed elementary logical conditions, also allows a nondeterministic operation on operators. For algebras of nondeterministic algorithms with closed logical conditions, the so-called S(H)-algebras, we construct a complete finite system of axioms with substitutions as the only inference rule. An algorithm is developed for the reduction of regular schemas in these algebras to canonical form, i.e., to representation in standard nondeterministic polynomials; as a consequence, we establish decidability of the problem of identities in these algebras.