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In this paper we complete the classification of the elliptic fibrations on K3surfaces which admit a non-symplectic involution acting trivially on theN\'eron--Severi group. We use the geometric method introduced by Oguiso andmoreover we provide a geometric construction of the fibrations classified. Ifthe non-symplectic involution fixes at least one curve of genus 1, we relateall the elliptic fibrations on the K3 surface with either elliptic fibrationsor generalized conic bundles on rational elliptic surfaces. This descriptionallows us to write the Weierstrass equations of the elliptic fibrations on theK3 surfaces explicitly and to study their specializations.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus