Suppose we are given a system:
$$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$ $$...$$ $$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$
And are interested in finding subspaces of the vector space that are invariant under this system, i.e. trajectories starting in these subspace remain in these subspace.
In other-words, we want to find subsets of the phase space $S$ such that for every trajectory $l(t) = <x_{1}(t),...,x_{n}(t)>$ if $l(0) \in S$ then $l(t) \in S$ for every $t$.
Obviously, limit cycles and fixed points and fixed lines are examples. But is there a general method for finding all such subspace?
Generally there is no one-size-fits-all method, but there are many techniques that can work under the right circumstances. For example, it is known that invariant manifolds (stable, unstable, center) corresponding to a fixed points are tangent to the invariant subspaces of the linearized system, and these can be sometimes extended to characterize the full invariant manifolds of the nonlinear system. If you can construct a Lyapunov function for your system, then level sets of that function are typically invariant.
The most general method I know if is a relatively recent development in terms of Koopman eigenfunctions. Consider an ODE $x' = f(x)$ defined on some manifold $\mathcal{M}$ as well as some sufficiently nice function space $\mathcal{V}(\mathcal{M})$. We then define a family of operators, known as Koopman operators, by the formula $$\mathcal{K}_t \phi = \phi\circ F_t~\forall\phi\in\mathcal{V}(\mathcal{M}),$$ where $F_t$ is the flow map induced by the ODE, i.e., $F_t(x_0) = x(t)$, where $x(t)$ solves the ODE with initial condition $x_0$. Under relatively mild conditions on the ODE and the function space one is consider, one can find eigenfunctions of the family of Koopman operators satisfying $$\mathcal{K}_t \psi_k = e^{\lambda_k t}\psi_k.$$ It turns out that under mild assumptions, the level sets of these eigenfunctions can generate invariant partitions of the state space, with the real components of the corresponding eigenvalues determining stability properties of some of the elements of the partition. See the cited papers for more details. Igor Mezic also has some lectures on this topic on his YouTube channel if you want more details.
Budišić, Marko; Mezić, Igor, Geometry of the ergodic quotient reveals coherent structures in flows, Physica D 241, No. 15, 1255-1269 (2012). ZBL1254.37010.
Mauroy, Alexandre; Mezić, Igor, Global stability analysis using the eigenfunctions of the Koopman operator, IEEE Trans. Autom. Control 61, No. 11, 3356-3369 (2016). ZBL1359.93372.