The author deals with a quasilinear system of the form \[ \dot x=Ax+f_1(x,y,\alpha), \qquad \dot y=By+f_2(x,y,\alpha), \] where \(x\) and \(y\) are, respectively, \(n\)- and \(m\)-dimensional vectors, \(A\), \(B\) are diagonal matrices, \(\alpha\) is \(d\)-dimensional parameter, and \(f_i(x,y,\alpha)=O(\|x\|^2 +\|y\|^2)\), \(i=1,2\), are continuous functions. The case is considered when for \(\alpha =0\) the trivial solution is unstable and for any \(\alpha\) the system possesses an invariant set. In polar coordinates \(x_j=r_j\exp(i\theta_j)\), \(y_k=\rho _k\exp(i\varphi_k)\), \(j=1,\ldots,n\), \(k=1,\ldots,m\), this set has the form \(A(\alpha)=\{r\in{\mathbb{R}}^{n}\), \(\rho \in{\mathbb{R}}^{m}:\|r\|+\|\rho\|=\kappa(\alpha)\}\) where \(\kappa(\alpha)\) is such a positive function that \(\kappa(\alpha)\to\infty\), \(\|\alpha\|\to\infty\). In terms of the matrix Lyapunov function \(V(r,\rho)= \sum_{i,j=1}^{2}u_{ij}(r,\rho)\eta_i\eta_j\) the author obtains sufficient conditions for the uniform asymptotic stability of the set \(A(\alpha)\). Here, \(u_{11}(r,\rho)= \sum_{s=1}^{n_1}\nu_sr_s^2, u_{22}(r,\rho)= \sum_{s=1}^{m_1}\mu_s\rho_s^2\)\(, \)\(u_{12}(r,\rho)=u_{21}(r,\rho)\) is a quadratic form, \(\nu_s\), \(\mu_s\), \(\eta_j\) are positive numbers.