The Riemann and Einstein-Weyl geometries in the theory of ordinary differential equations their applications and all that

Some properties of the 4-dimensional 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) are considered. Three-dimensional Einstein-Weyl spaces connected with dual equations b" = (a, b, b') where the function g(a, b, b') satisfies the partial differential equation g(aacc) + 2cg(abcc) + 2gg(accc) + c(2)g(bbcc) + 2cgg(bccc) + g(2)g(cccc) + (g(a) + Cg(b))g(ccc) - 4g(abc) - 4cg(bbc) - cg(c)g(bcc) - 3gg(bcc) - g(c)g(acc) + 4g(c)g(bc) - 3g(b)g(cc) + 6g(bb) = 0 are also investigated. The theory of invariants for second order ODE's is applied to the study of nonlinear dynamical systems dependent on a set of parameters.