Let
X
X
be a smooth projective variety over the complex numbers and
S
(
d
)
S(d)
the scheme parametrizing
d
d
-dimensional Lie subalgebras of
H
0
(
X
,
T
X
)
H^0(X,\mathcal {T}X)
. This article is dedicated to the study of the geometry of the moduli space
Inv
\text {Inv}
of involutive distributions on
X
X
around the points
F
∈
Inv
\mathcal {F}\in \text {Inv}
which are induced by Lie group actions. For every
g
∈
S
(
d
)
\mathfrak {g}\in S(d)
one can consider the corresponding element
F
(
g
)
∈
Inv
\mathcal {F}(\mathfrak {g})\in \text {Inv}
, whose generic leaf coincides with an orbit of the action of
exp
(
g
)
\exp (\mathfrak {g})
on
X
X
. We show that under mild hypotheses, after taking a stratification
∐
i
S
(
d
)
i
→
S
(
d
)
\coprod _i S(d)_i\to S(d)
this assignment yields an isomorphism
ϕ
:
∐
i
S
(
d
)
i
→
Inv
\phi :\coprod _i S(d)_i\to \text {Inv}
locally around
g
\mathfrak {g}
and
F
(
g
)
\mathcal {F}(\mathfrak {g})
. This gives a common explanation for many results appearing independently in the literature. We also construct new stable families of foliations which are induced by Lie group actions.