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In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved.

We present a discrete multiscale expansion of the lattice potential Korteweg-de Vries (lpKdV) equation on functions of infinite order of slow-varyness. To do so we introduce a formal expansion of the shift operator on many lattices holding at all orders. The lowest secularity condition from the expansion of the lpKdV equation gives a nonlinear lattice equation, depending on shifts of all orders, of the form of the nonlinear Schr\"odinger (NLS) equation, Comment: 9 pages, submitted to Journ. Phys. A

We discuss the non autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its B\"acklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the $H^4$ trapezoidal and the $H^6$ families are linearizable and in a few examples we show how we can effectively linearize them.

We provide conditions for a lattice scheme defined on a four points lattice to be linearizable by a point transformation. We apply the obtained conditions to a symmetry preserving difference scheme for the potential Burgers introduced by Dorodnitsyn \cite{db} and show that it is not linearizable.

"We show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole–Hopf transformation. If the equation is linearizable, then the symmetry provides the linearizing transformation. At the end, we present a few examples of applications for equations defined on four lattice points"

We use the conditions of linearizability by point and by Hopf–Cole transformations, introduced recently by the authors, to classify all real multilinear equations on a square lattice with at most quadratic nonlinearity. We find that, up to a linear transformation of the dependent variable and an exchange of the independent variables, only two equations in this class are linearizable by point transformations and none by Hopf–Cole transformations.