In this paper, we consider the persistence of completely degenerate lower-dimensional invariant tori of the following reversible system
{
x
˙
=
ω
+
Q
(
x
)
z
+
ϵ
P
1
(
x
,
z
,
ϵ
)
,
z
˙
=
H
(
z
)
+
ϵ
P
2
(
x
,
z
,
ϵ
)
,
\begin{eqnarray*} \left \{ \begin {array}{l} \dot x=\omega +\mathcal {Q}(x)z+\epsilon \mathcal {P}^1(x,z,\epsilon ),\\ \dot z=\mathcal {H}(z)+\epsilon \mathcal P^2(x,z,\epsilon ),\\ \end{array} \right . \end{eqnarray*}
where
(
x
,
z
)
=
(
x
,
y
,
u
,
v
)
∈
T
n
×
R
m
×
R
×
R
(x,z)=(x,y,u,v)\in \mathbb {T}^n\times \mathbb {R}^m\times \mathbb {R}\times \mathbb {R}
with
m
≥
n
+
2
m\geq n+2
,
H
(
z
)
=
(
0
,
v
2
p
+
1
+
y
m
l
,
u
y
m
−
1
q
)
T
\mathcal {H}(z)=(0,v^{2p+1}+y_m^{l},uy_{m-1}^q)^T
with
y
=
(
y
1
,
⋯
,
y
m
−
1
,
y
m
)
y=(y_1,\cdots ,y_{m-1},y_{m})
,
p
,
q
≥
0
p,q\geq 0
,
l
>
0
l>0
are integers, the involution
G
G
is
(
x
,
y
,
u
,
v
)
→
(
−
x
,
y
,
−
u
,
v
)
(x,y,u,v)\rightarrow (-x,y,-u,v)
,
Q
(
x
)
\mathcal {Q}(x)
is a
n
×
m
n\times m
matrix function,
ω
\omega
is a Diophantine frequency,
ϵ
\epsilon
is a small positive parameter and
ϵ
P
1
,
ϵ
P
2
\epsilon \mathcal {P}^1,\,\,\epsilon \mathcal {P}^2
are analytic perturbation terms. By the Kolmogorov-Arnold-Moser method, we prove that for sufficiently small
ϵ
\epsilon
the above reversible system admits lower-dimensional invariant tori with prescribed frequency
ω
\omega
if average of a part of
Q
(
x
)
\mathcal {Q}(x)
is non-singular. This should be the first persistence result of lower-dimensional invariant tori in completely degenerate reversible systems.