Authors: Takayuki Koike ¹; Takato Uehara ²

• 1 Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Osaka 558-8585, Japan
• 2 Department of Mathematics, Faculty of Science, Okayama University, 1-1-1, Tsushimanaka, Okayama, 700-8530, Japan

We construct a non-Kummer projective K3 surface $X$ which admits compact Levi-flats by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective plane at nine general points.