<p>We consider the topic of integrability, i.e. when an analytical solution to a system of differential equations can be found. It is not always possible to find such solutions, and one approach to simplify the problem is to search for specific parameter values at which the system is locally integrable. This leads to the hypothesis that we need local integrability at each point of the phase space to have integrability of the system overall. We explore this hypothesis experimentally for a two-dimensional polynomial autonomous ODE system. In the case when the system under study depends on parameters, then Bruno’s normal form method allows us to reduce the requirements of this hypothesis toa system of algebraic equations for the parameters of the system: by solving these equations, we find integrable cases. The method works in cases of resonance in the linear part of the system. We applied this method to study the Bautin system, a two-dimensional ODE with quadratic nonlinearity on the right-hand sides. For resonance cases 1:1and 1:2, a few dozen first integrals were found and are presented. We also report that the algebraic system of parameters obtained by combining systems of equations for resonances 1:1, 1:2 and 1:3 has 11 solutions, each of which corresponds to an integrable case of the Bautin system of a general (non-resonant) type. This demonstrates the possibility of using this method outside resonant regions.</p>