  A stratified Lie system is a nonautonomous system of first-order ordinary
differential equations on a manifold $M$ described by a $t$-dependent vector
field $X=\sum_{\alpha=1}^rg_\alpha X_\alpha$, where $X_1,\ldots,X_r$ are vector
fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the
strata of a stratification $\mathcal{F}$ of $M$ while
$g_1,\ldots,g_r:\mathbb{R}\times M\rightarrow \mathbb{R}$ are functions
depending on $t$ that are constant along integral curves of $X_1,\ldots,X_r$
for each fixed $t$. We analyse the particular solutions of stratified Lie
systems and how their properties can be obtained as generalisations of those of
Lie systems. We illustrate our results by studying Lax pairs and a class of
$t$-dependent Hamiltonian systems. We study stratified Lie systems with
compatible geometric structures. In particular, a class of stratified Lie
systems on Lie algebras are studied via Poisson structures induced by
$r$-matrices.
