  We study special triple covers $f\colon T \to S$ of algebraic surfaces, where
the Tschirnhausen bundle $\mathcal E = \left(f_*\mathcal O_T/\mathcal
O_S\right)^\vee$ is a quotient of a split rank three vector bundle, and we
provide several necessary and sufficient criteria for the existence.
  As an application, we give a complete classification of special triple
planes, finding among others two nice families of K3 surfaces.
