  Let $Y$ be the complement of a plane quartic curve $D$ defined over a number
field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is
a generic smooth quartic curve, by showing that its integral points are
confined in a curve except for a finite number of exceptions. The required
finiteness will be obtained by reducing it to the Shafarevich conjecture for K3
surfaces. Some variants of our method confirm the same conjecture when $D$ is a
reducible generic quartic curve which consists of four lines, two lines and a
conic, or two conics.
