Singular perturbed integro-differential Volterra equation and Drazin's inverse singular matrices

We consider integro-differential system of the form A (μ) frac(d x, d t) = B (μ) x + G (t, μ, x) + ∫₀^t K (t, s, μ) φ (x (s, μ)) d s, where the coefficients A (μ), B (μ), G (t, μ, x) and K (t, s, μ) continuously depend on a vector parameter μ, and the matrix A (μ) is singular at μ = 0. For analysis of this system, a method based on its reduction to a countable singular system of ordinary differential equations is proposed. The finite sections method and an application of Drazin's inverse matrices to the integration of linear singular systems are used.