Differential schemes for the ordinary differential equations defining a projective correspondence between layers

It is well known that there are remarkable differential equations which can be integrated in CAS, but there are several inequivalent approaches for description of these differential equations. In our work we want to discuss remarkable differential equations in another sense: for these equations there exist finite difference schemes which conserve algebraic properties of solutions exactly. It should be noted that this class of differential equations coincides with the class introduced by Painlevé. In terms of Cauchy problem a differential equation of this class defines an algebraic correspondence between initial and terminal values. For example Riccati equation $y'=p(x)y^2+q(x)y+r(x)$ defines one-to-one correspondence between initial and terminal values of $y$ on projective line. However, standard finite difference schemes do not conserve this algebraic property of exact solution. Furthermore, the scheme, which defines one-to-one correspondence between layers, truly describes solution not only before but also after mobile singularities and conserves algebraic properties of equations like the anharmonic ratio. After necessary introduction (sections 1 and 2) we describe such one-to-one scheme for Riccati equation and prove its properties mentioned above.