  Using advantages of nonstandard computational techniques based on the
light-cone variables, we explicitly find the algebra of generalized symmetries
of the (1+1)-dimensional Klein-Gordon equation. This allows us to describe this
algebra in terms of the universal enveloping algebra of the essential Lie
invariance algebra of the Klein-Gordon equation. Then we single out variational
symmetries of the corresponding Lagrangian and compute the space of local
conservation laws of this equation, which turns out to be generated, up to the
action of generalized symmetries, by a single first-order conservation law.
Moreover, for every conservation law we find a conserved current of minimal
order that is contained in this conservation law.
