This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl.387 (2012), 776–790].

We give a natural notion of non-degeneracy for singular points of integrable non-Hamiltonian systems, and show that such non-degenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case, and prove this conjecture for systems of type $(n, 0)$, i.e. $n$ commuting smooth vector fields on an $n$-manifold.

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.