  For partial differential equations (PDEs) that have $n\geq2$ independent
variables and a symmetry algebra of dimension at least $n-1$, an explicit
algorithmic method is presented for finding all symmetry-invariant conservation
laws that will reduce to first integrals for the ordinary differential equation
(ODE) describing symmetry-invariant solutions of the PDE. This significantly
generalizes the double reduction method known in the literature. Moreover, the
condition of symmetry-invariance of a conservation law is formulated in an
improved way by using multipliers, thereby allowing symmetry-invariant
conservation laws to be obtained directly, without the need to first find
conservation laws and then check their invariance. This cuts down considerably
the number and complexity of computational steps involved in the reduction
method. If the space of symmetry-invariant conservation laws has dimension
$m\geq 1$, then the method yields $m$ first integrals along with a check of
which ones are non-trivial via their multipliers. Several examples of
interesting symmetry reductions are considered: travelling waves and similarity
solutions in $1+1$ dimensions; line travelling waves, line similarity
solutions, and similarity travelling waves in $2+1$ dimensions; rotationally
symmetric similarity solutions in $n+1$ dimensions. In addition, examples of
nonlinear PDEs for which the method yields the explicit general solution for
symmetry-invariant solutions are shown.
