In this paper, we consider the existence and multiplicity of normalized solutions for the following fractional Schrödinger system with Hardy critical coupled term:
{
(
−
Δ
)
s
u
+
λ
1
u
=
|
u
|
p
−
2
u
+
α
ν
2
s
,
τ
*
|
u
|
α
−
2
|
v
|
β
u
|
x
|
τ
,
in
ℝ
N
,
(
−
Δ
)
s
v
+
λ
2
v
=
|
v
|
p
−
2
v
+
β
ν
2
s
,
τ
*
|
u
|
α
|
v
|
β
−
2
v
|
x
|
τ
,
in
ℝ
N
,
∫
ℝ
N
u
2
d
x
=
a
2
,
∫
ℝ
N
v
2
d
x
=
b
2
,
where
N
≥
3
s
,
s
∈
(
0
,
1
)
,
τ
∈
(
0
,
2
s
)
,
a
,
b
>
0
,
ν
>
0
, and the exponents
α
,
β
,
p
satisfy
α
>
1
,
β
>
1
,
α
+
β
=
2
s
,
τ
*
:
=
2
(
N
−
τ
)
N
−
2
s
,
2
<
p
<
2
s
*
:
=
2
N
N
−
2
s
the parameters
λ
1
,
λ
2
∈
ℝ
appear as unknown Lagrange multipliers. In the case of
2
<
p
<
2
+
4
s
N
, we get the existence of a normalized ground state solution and a mountain-pass type normalized solution by employing the minimax principle, the Pohozaev manifold analysis, homotopy-stable family and the mountain pass theorem. In the case of
2
+
4
s
N
≤
p
<
2
s
*
, we obtain the existence of a mountain-pass type normalized solution by the Pohozaev manifold analysis and the mountain pass theorem.