Generalized-conjugate transformations and integrability in quadrature of some classes of systems

Uniform two-dimensional systems of linear differential equations with variable coefficients, as well as the second order equations with variable coefficients are considered. A linear transformation of the desired functions is introduced, the coefficients of which satisfy a homogeneous linear system of differential equations that are generalized-associated to the source system (or one second-order equation), which contains some arbitrary function. In the case when this function can be selected in such a way that the generalized-conjugate system (equation) has a solution in quadratures or, in particular, in an obvious analytical form, the source system (or equation) is also integrated in squares. A number of examples of cases of integrability in the quadratures of systems and equations are given