  The paper is devoted to the conjecture that an equation is Darboux integrable
if and only if it possesses symmetries depending on arbitrary functions. We
note that results of previous works together prove this conjecture for scalar
partial differential equations of the form $u_{xy}=F(x,y,u,u_x,u_y)$. For
autonomous semi-discrete and discrete analogues of these equations we prove
that the sequence of Laplace invariants is terminated by zero for an equation
if this equation admits an operator mapping any function of one independent
variable into a symmetry of the equation. The vanishing of an Laplace invariant
allows us to construct a formal integral, i.e. an operator that maps symmetries
into integrals (including, generally speaking, trivial integrals). This and
results of previous works together prove a `formal' version of the
aforementioned conjecture in the semi-discrete and pure discrete cases.
