K-trivial structures on Fano complete intersections

It is proven that any structure of a fibre space into varieties of Kodaira dimension zero on a generic Fano complete intersection of index one and dimension $M$ in ${\mathbb{P}}^{M+k}$ for $M\ge 2k+1$ is a pencil of hyperplane sections. We also describe $K$-trivial structures on varieties with a pencil of Fano complete intersections.