A criterion in the spirit of Rosenlicht is given, on the rational function
f
(
x
)
f(x)
, for when the planar vector field
{
y
′
=
x
y
x
′
=
f
(
x
)
}
\left \{\begin {smallmatrix} y’=xy\\ x’=f(x) \end {smallmatrix}\right \}
admits a pair of algebraically independent first integrals over some extension of the base field. This proceeds from model-theoretic considerations by working in the theory of differentially closed fields of characteristic zero and asking: If
D
⊆
A
1
D\subseteq \mathbb A^1
is a strongly minimal set internal to the constants, when is
log
δ
−
1
(
D
)
\operatorname {log}_{\delta }^{-1}(D)
, the pullback of
D
D
under the logarithmic derivative, itself internal to the constants?