Spray simulation and point symmetries of a spray (Q1603031)

One considers a spray \(f\) on a smooth manifold \(M\) given in local coordinates by \(d^2x^i/dt^2 = f^i(x^j, dx^j/dt)\) with \(f^i = - \Gamma ^i_{jk}(x,y)y^jy^k\), where \(\Gamma ^i_{jk}\) are homogeneous functions of degree \(0\) in \(y^h = dx^h/dt\) for indices ranging over \({1,2,\dots , n-1 = \dim M }\). One associates to \(f\) a linear connection in \(M\times R\) and studies the infinitesimal affine transformations of it. A general form of point affine symmetries is provided and a classification of 2-dimensional affinely connected manifolds by algebras of proper point affine symmetries is given. This is based on a paper by \textit{J. Levine} [Ann. Math. (2) 52, 465-477 (1950; Zbl 0038.34603)].