  We show how the classical string dynamics in $D$-dimensional curved
background can be reduced to the dynamics of a massless particle constrained on
a certain surface whenever there exists at least one Killing vector for the
background metric. Then we obtain a number of sufficient conditions, which
ensure the existence of exact solutions to the equations of motion and
constraints. The results are also relevant to the null string case. Finally, we
illustrate our considerations with an explicit example in four dimensions.
