In 1945 A. Einstein [6] and E. Schrödinger [11] started form a generalized Riemann space, that is, a space $\mathrm M$ associated with a nonsym-metric tensor $G_{ij}(x)$ and desired to find the set of all linear connections $\Gamma^i_{jk}(x)$ compatible with such a metric: $G_{ij/k}=0$ (see also [l]). The geometry of this space $(\mathrm M,G_{ij})$ is called \emph{the Einstein-Schrödinger's geometry} [3, 4]. The purpose of this paper is to discuss a nonsymmotric tensor field $G_{ij}(x,y^{(1)},y^{(2)})$, where $(x,y^{(1)},y^{(2)})$ is a point of the osculator bundle of the second order $(Osc^2M,\pi,M)$ and to obtain the results for the Einstein-Schrödinger's geometry of the order two in a natural case. The fundamental notions and notations concerning the osculator bundle of the order two are in the papers [2, 5, 7, 8]. We give, shortly, at the beginning the notations used.