Bivariate, hyperbolic third-order linear partial differential operators under the gauge transformations L → g(x,y){sup}(-1) ○ L ○ g(x,y) are considered. The existence of a factorization, the existence of a factorization that extends a given factorization of the symbol of the operator are expressed in terms of the invariants of some known generating set of invariants. The operation of taking the formal adjoint can be also defined for equivalent classes of LPDOs, and explicit formulae defining this operation in the space invariants were obtained.
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