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The paper is devoted to the conjecture that an equation is Darboux integrableif and only if it possesses symmetries depending on arbitrary functions. Wenote that results of previous works together prove this conjecture for scalarpartial differential equations of the form $u_{xy}=F(x,y,u,u_x,u_y)$. Forautonomous semi-discrete and discrete analogues of these equations we provethat the sequence of Laplace invariants is terminated by zero for an equationif this equation admits an operator mapping any function of one independentvariable into a symmetry of the equation. The vanishing of an Laplace invariantallows us to construct a formal integral, i.e. an operator that maps symmetriesinto integrals (including, generally speaking, trivial integrals). This andresults of previous works together prove a `formal' version of theaforementioned conjecture in the semi-discrete and pure discrete cases.

On relationships between symmetries depending on arbitrary functions and integrals of discrete equations