Let
k
k
be a field of characteristic
p
>
0
p > 0
and
D
≠
0
D \ne 0
a family of
k
k
-derivations of
k
[
x
,
y
]
k[x,y]
. We prove that
k
[
x
,
y
]
D
k{[x,y]^D}
,the ring of constants with respect to
D
D
, is a free
k
[
x
p
,
y
p
]
k[{x^p},{y^p}]
-module of rank
p
p
or 1 and
k
[
x
,
y
]
D
=
k
[
x
p
,
y
p
,
f
1
,
…
,
f
p
−
1
]
k{[x,y]^D} = k[{x^p},{y^p},{f_1}, \ldots ,{f_{p - 1}}]
for some
f
1
,
…
,
f
p
−
1
∈
k
[
x
,
y
]
D
{f_1}, \ldots ,{f_{p - 1}} \in k{[x,y]^D}
.