The authors consider the equation \[ \dot x = f(t,x),\quad x(t_0) = x_0,\tag{1} \] where \(f\in C(\mathbb{R}^+\times D,\mathbb{R}^s)\) is locally Lipschitzian in \(x\), \(f(t,0)\equiv 0\), \(D\subset \mathbb{R}^s\), and the perturbed equation \[ \dot x = g(t,x,\lambda),\quad x(t_0) = x_0,\tag{2} \] where \(g: \mathbb{R}^+\times D\times\Lambda\to \mathbb{R}^s\) be a mapping such that: (i) for each \(\lambda\in\Lambda\) the function \(g(\cdot,\cdot,\lambda)\) is locally Lipschitzian in \(x\), and (ii) \(g(\cdot,\cdot,\lambda) = f\) if and only if \(\lambda = 0\). Let \({\mathcal U} = \{g(\cdot,\cdot,\lambda)-f: \lambda\in\Lambda\}\) be the set of perturbations corresponding to \(\Lambda\) and denote by \({\mathcal U}^*\) the set \({\mathcal U}-\{0\}\). Let \(g(t,0,\lambda) = 0\) and (2) admits the solution \(x(t)=0\). The solution \(x(t)=0\) to the unperturbed equation (1) is said to be \({\mathcal U}^*\)-secularly stable if \(x(t)\equiv 0\) is a uniformly stable solution to (2) for any \(\lambda \in\Lambda-\{0\}\). The theorems on the secular stability are proved and applied to holonomic systems.