A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under

A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$

A set $A \subset \mathbb{R}^2$ containing more than one point that is not invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$

Are these possible? if so, could you show me examples?