  We study the real-analytic continuation of local real-analytic solutions to
the Cauchy problems of quasi-linear partial differential equations of first
order for a scalar function. By making use of the first integrals of the
characteristic vector field and the implicit function theorem we determine the
maximal domain of the analytic extension of a local solution as a single-valued
function. We present some examples including the scalar conservation laws that
admit global first integrals so that our method is applicable.
