  In this paper we complete the classification of the elliptic fibrations on K3
surfaces which admit a non-symplectic involution acting trivially on the
N\'eron--Severi group. We use the geometric method introduced by Oguiso and
moreover we provide a geometric construction of the fibrations classified. If
the non-symplectic involution fixes at least one curve of genus 1, we relate
all the elliptic fibrations on the K3 surface with either elliptic fibrations
or generalized conic bundles on rational elliptic surfaces. This description
allows us to write the Weierstrass equations of the elliptic fibrations on the
K3 surfaces explicitly and to study their specializations.
