For a real analytic complex vector field
L
L
, in an open set of
R
2
\mathbb {R}^2
, with local first integrals that are open maps, we attach a number
μ
≥
1
\mu \ge 1
(obtained through Łojasiewicz inequalities) and show that the equation
L
u
=
f
Lu=f
has bounded solution when
f
∈
L
p
f\in L^p
with
p
>
1
+
μ
p>1+\mu
. We also establish a similarity principle between the bounded solutions of the equation
L
u
=
A
u
+
B
u
¯
Lu=Au+B\overline {u}
(with
A
,
B
∈
L
p
A,B\in L^p
) and holomorphic functions.