Reference request: how to check whether a set is invariant for a second order dynamical system?

Question

I am looking for some examples where invariant set is proved for second order systems

For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$

In general, what is the procedure to prove that the solution is invariant with respect to

1. A "regular" set i.e. $D = \{(x_1, x_2)|x_1^2+ x_2^2 = 1\}$
2. A set with corners i.e. $D = \{(x_1, x_2)|x_1+ x_2 = 1\}$