In the talk autonomous systems of differential equations with rational right part will be investigated and the following questions will be considered:

• methods of finding algebraic integrals and integral varieties and implementing them in computer algebra systems (Sage),
• constructing of conservative difference schemes, which preserve exactly integrals or integral varieties and fundamental obstacles for constructing explicit conservative schemes
• organization of calculation based on implicit differential schemes

We will present:

• package for calculation of integral and Darboux polynomial (based on Lagutinski method) in Sage,
• package for Butcher varieties for simple and symplectic Runge-Kutta scheme in Sage,
• results of numerical experiments obtained by applying the midpoint scheme to an investigation study of linear and elliptical oscillators.

The report will be accompanied with experiments conducted in Sage.