Bimetric physical structures of rank $(n+1,2)$

For $s\ge1$ and $n\ge m\ge1$, we give a concise definition for an s-metric physical structure of rank $(n+1,m+1)$ determined by an $s$-component function $f=(f^1,\ldots,f^s)$ on sets $\mathfrak{M}$ and $\mathfrak{N}$ (an $sm$-dimensional manifold and an $sn$-dimensional manifold). The function $f$ is denned on $\mathfrak{G}_f\subset\mathfrak{M}\times\mathfrak{N}$ and carries each pair in $\mathfrak{G}_f$ into $s$ numbers; $f$ is called an $s$-metric. We prove that bimetric $(s=2)$ physical structures of rank $(n+1,2)$ exist only if $n=1,2,3,4$. Explicit coordinate expressions of all (up to equivalence) two-metrics are provided. The study is based on the group properties of physical structures which were earlier studied by the author and on a complete classification of finite-dimensional Lie groups of plane transformations. Some of the two-metrics obtained specify natural binary operations of addition and multiplication in $\mathbb{R}^2$ which can, in particular, be used to define three types of two-dimensional complex numbers (ordinary, dual, and double).