I know that a Lyapunov stable equilibrium doesn't have to be isolated, but cannot think of a system of equations that satisfy this.
I also know that an equilibrium $x^*$ is Lyapunov stable: if $\forall\epsilon>0$ $\exists\delta(\epsilon)>0$ such that when $|x-x^*|<\delta$ we have $|\phi(x,t)-x^*|<\epsilon$ for all $t\geq0$.
and $x^*$ is isolated if if there's a neighbourhood around it that has no other equilibria.