i.e. then these equations have the unique solution: At this step, for each root of the polynomial a corresponding set of coefficients is computed. Step 3. Check factorization condition (which is the last of the initial 6 equations) written in the known variables and ): If the operator is factorizable and explicit form for the factorization coefficients is given above. Operator of order 3 Consider an operator with smooth coefficients and look for a factorization Similar to the case of the operator the conditions of factorization are described by the following system: with and again i.e. and three-step procedure yields: At the first step, the roots of a cubic polynomial have to be found. Again denotes a root and first four coefficients are At the second step, a linear system of three algebraic equations has to be solved: At the third step, two algebraic conditions have to be checked. Operator of order Invariant Formulation Definition The operators , are called equivalent if there is a gauge transformation that takes one to the other: BK-factorization is then pure algebraic procedure which allows to construct explicitly a factorization of an arbitrary order LPDO in the form with first-order operator where is an arbitrary simple root of the characteristic polynomial Factorization is possible then for each simple root iff for for for and so on. All functions are known functions, for instance, and so on. Theorem All functions are invariants under gauge transformations. Definition Invariants are called generalized invariants of a bivariate operator of arbitrary order. In particular case of the bivariate hyperbolic operator its generalized invariants coincide with Laplace invariants (see Laplace invariant). Corollary If an operator is factorizable, then all operators equivalent to it, are also factorizable. Equivalent operators are easy to compute: and so on. Some example are given below: Transpose Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need right factors and BK-factorization constructs left factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the transpose of that operator. Definition The transpose of an operator is defined as and the identity implies that Now the coefficients are with a standard convention for binomial coefficients in several variables (see Binomial coefficient), e.g. in two variables In particular,