The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation
A
(
x
)
y
′
′
+
2
B
(
x
)
y
′
+
C
(
x
)
y
=
0
,
\begin{equation*} A(x) y^{\prime \prime } + 2 B(x) y’ + C(x) y = 0, \end{equation*}
where
A
(
x
)
,
B
(
x
)
A(x), B(x)
and
C
(
x
)
C(x)
are polynomials of degree
p
+
1
,
p
p+1, p
and
p
−
1
p-1
, is under discussion. We concentrate on the case when
A
(
x
)
A(x)
has only real zeros
a
j
a_{j}
and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients
r
j
r_{j}
in the partial fraction decomposition
B
(
x
)
/
A
(
x
)
=
∑
j
=
0
p
r
j
/
(
x
−
a
j
)
B(x)/A(x) = \sum _{j=0}^{p} r_{j}/(x-a_{j})
, we allow the presence of both positive and negative coefficients
r
j
r_{j}
. The corresponding electrostatic interpretation of the zeros of the solution
y
(
x
)
y(x)
as points of equilibrium in an electrostatic field generated by charges
r
j
r_{j}
at
a
j
a_{j}
is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.