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Recently, Lembert, Gilson et al proposed a lucid and systematic approach toobtain bilinear B\"{a}cklund transformations and Lax pairs forconstant-coefficient soliton equations based on the use of binary Bellpolynomials. In this paper, we would like to further develop this method withnew applications. We extend this method to systematically investigate completeintegrability of nonisospectral and variable-coefficient equations. Inaddiction, a method is described for deriving infinite conservation laws ofnonlinear evolution equations based on the use of binary Bell polynomials. Allconserved density and flux are given by explicit recursion formulas. By takingvariable-coefficient KdV and KP equations as illustrative examples, theirbilinear formulism, bilinear B\"{a}cklund transformations, Lax pairs, Darbouxcovariant Lax pairs and conservation laws are obtained in a quick and naturalmanner. In conclusion, though the coefficient functions have influences on avariable-coefficient nonlinear equation, under certain constrains the equationturn out to be also completely integrable, which leads us to a canonicalinterpretation of their $N$-soliton solutions in theory.

Binary Bell polynomials approach to the integrability of nonisospectral and variable-coefficient nonlinear equations