  Some properties of the 4-dim Riemannian spaces with metrics $$
ds^2=2(za_3-ta_4)dx^2+4(za_2-ta_3)dxdy+2(za_1-ta_2)dy^2+2dxdz+2dydt $$
associated with the second order nonlinear differential equations $$
y''+a_{1}(x,y){y'}^3+3a_{2}(x,y){y'}^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 $$ with
arbitrary coefficients $a_{i}(x,y)$ and 3-dim Einstein-Weyl spaces connected
with dual equations $$ b''=g(a,b,b') $$ where the function $g(a,b,b')$
satisfied the partial differential equation $$
g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}+
g^2g_{cccc}+(g_a+cg_b)g_{ccc}-4g_{abc}- $$ $$
  -4cg_{bbc} -cg_{c}g_{bcc}- 3gg_{bcc}-g_cg_{acc}+
4g_cg_{bc}-3g_bg_{cc}+6g_{bb} =0 $$ are considered.
