We consider orthogonal polynomials in two variables whose derivatives with respect to
x
x
are orthogonal. We show that they satisfy a system of partial differential equations of the form
α
(
x
,
y
)
∂
x
2
U
→
n
+
β
(
x
,
y
)
∂
x
U
→
n
=
Λ
n
U
→
n
,
\begin{equation*} \alpha (x,y)\partial _{x}^{2}\overrightarrow {U}\!_{n}+\beta (x,y)\partial _{x} \overrightarrow {U}\!_{n}=\Lambda _{n}\overrightarrow {U}\!_{n}, \end{equation*}
where
deg
α
≤
2
\deg \alpha \leq 2
,
deg
β
≤
1
\deg \beta \leq 1
,
U
→
n
=
(
U
n
0
,
U
n
−
1
,
1
,
⋯
,
U
0
n
)
\overrightarrow {U} _{n}=(U_{n0},U_{n-1,1},\cdots ,U_{0n})
is a vector of polynomials in
x
x
and
y
y
for
n
≥
0
n\geq 0
, and
Λ
n
\Lambda _{n}
is an eigenvalue matrix of order
(
n
+
1
)
×
(
n
+
1
)
(n+1)\times (n+1)
for
n
≥
0
n\geq 0
. Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran’s.