There is proposed an asymptotic expansion for the fundamental system of solutions of a linear singularly perturbed system of \(m\)-order differential equations with degenerate principal matrix at the higher order derivatives. The considered system of differential equation is of the form: \[ \sum_{i=0}^m \varepsilon^{ih}A_i(t,\varepsilon)\frac{d^ix}{dt^i}=0, \] where \(x(t,\varepsilon)\) is the required \(n\)-dimensional vector, \(A_i(t,\varepsilon), i=\overline{0,m}\), are real or complex valued \(n\times n\)-matrices, \(\varepsilon\in(0,\varepsilon_0]\) is a small real parameter and \(h\in {\mathbb N}, t\in [0,T]\). The case when the corresponding characteristic polynomial has a multiple spectrum is considered. In order to study the asymptotics of linearly independent solutions of the above system, the theory of polynomial matrix pencils is used. First, the main theorem specifying the form of formal solutions of the considered system in the case of multiple roots of the characteristic equation is proved. The proof of this theorem provides an algorithm used to find the coefficients of the corresponding formal expansions. Second, conditions under which the constructed formal solutions have the asymptotic character are formulated and the corresponding asymptotic estimates are presented.