  Smooth primitively polarized $\mathrm{K3}$ surfaces of genus 36 are studied.
It is proved that all such surfaces $S$, for which there exists an embedding
$\mathrm{R} \hookrightarrow \mathrm{Pic}(S)$ of some special lattice
$\mathrm{R}$ of rank 2, are parameterized up to an isomorphism by some
18-dimensional unirational algebraic variety. More precisely, it is shown that
a general $S$ is an anticanonical section of a (unique) Fano 3-fold with
canonical Gorenstein singularities.
