(Debug) In[]:=
H2[x_,y_]=
bCc(b(xα1+yγ1+δ1)+Cc(xα2+yγ2+δ2))
-1+b(xα2+yγ2+δ2)

-
2
b
-
2
Cc
(1-b(xα2+yγ2+δ2))
2
b
(1+Cc(xα1+yγ1+δ1)-b(xα2+yγ2+δ2))
;
​
(Debug) In[]:=
EE1=
bCc(b(0.5155229767052963`γ1+δ1)+Cc(0.5155229767052963`γ2+δ2))
-1+0.5155229767052963`bγ2+bδ2

-
2
b
-
2
Cc
(1-0.5155229767052963`bγ2-bδ2)
2
b
(1-0.5155229767052963`bγ2+Cc(0.5155229767052963`γ1+δ1)-bδ2)
-
bCc(b(7.131535846824116`γ1+δ1)+Cc(7.131535846824116`γ2+δ2))
-1+7.131535846824116`bγ2+bδ2

-
2
b
-
2
Cc
(1-7.131535846824116`bγ2-bδ2)
2
b
(1-7.131535846824116`bγ2+Cc(7.131535846824116`γ1+δ1)-bδ2)
;
(Debug) In[]:=
EE2=
bCc(b(1.1278966500259764`γ1+δ1)+Cc(1.1278966500259764`γ2+δ2))
-1+1.1278966500259764`bγ2+bδ2

-
2
b
-
2
Cc
(1-1.1278966500259764`bγ2-bδ2)
2
b
(1-1.1278966500259764`bγ2+Cc(1.1278966500259764`γ1+δ1)-bδ2)
-
bCc(b(6.519162173503435`γ1+δ1)+Cc(6.519162173503435`γ2+δ2))
-1+6.519162173503435`bγ2+bδ2

-
2
b
-
2
Cc
(1-6.519162173503435`bγ2-bδ2)
2
b
(1-6.519162173503435`bγ2+Cc(6.519162173503435`γ1+δ1)-bδ2)
;
​
(Debug) In[]:=
EE3=
bCc(b(1.928809010775717`γ1+δ1)+Cc(1.928809010775717`γ2+δ2))
-1+1.928809010775717`bγ2+bδ2

-
2
b
-
2
Cc
(1-1.928809010775717`bγ2-bδ2)
2
b
(1-1.928809010775717`bγ2+Cc(1.928809010775717`γ1+δ1)-bδ2)
-
bCc(b(5.718249812753696`γ1+δ1)+Cc(5.718249812753696`γ2+δ2))
-1+5.718249812753696`bγ2+bδ2

-
2
b
-
2
Cc
(1-5.718249812753696`bγ2-bδ2)
2
b
(1-5.718249812753696`bγ2+Cc(5.718249812753696`γ1+δ1)-bδ2)
;
​
(Debug) In[]:=
g=({EE1,EE2,EE3}/.{Cc1,b1,δ2->0.8})//FullSimplify
(Debug) Out[]=
-
0.8+7.13154γ1+7.13154γ2+δ1
-0.2+7.13154γ2

(0.2+7.13154γ1-7.13154γ2+δ1)
2
(0.2-7.13154γ2)
+
0.8+0.515523γ1+0.515523γ2+δ1
-0.2+0.515523γ2

(0.2+0.515523γ1-0.515523γ2+δ1)
2
(0.2-0.515523γ2)
,-
0.8+6.51916γ1+6.51916γ2+δ1
-0.2+6.51916γ2

(0.2+6.51916γ1-6.51916γ2+δ1)
2
(0.2-6.51916γ2)
+
0.8+1.1279γ1+1.1279γ2+δ1
-0.2+1.1279γ2

(0.2+1.1279γ1-1.1279γ2+δ1)
2
(0.2-1.1279γ2)
,-
0.8+5.71825γ1+5.71825γ2+δ1
-0.2+5.71825γ2

(0.2+5.71825γ1-5.71825γ2+δ1)
2
(0.2-5.71825γ2)
+
0.8+1.92881γ1+1.92881γ2+δ1
-0.2+1.92881γ2

(0.2+1.92881γ1-1.92881γ2+δ1)
2
(0.2-1.92881γ2)

I want to solve the following system but I can' t (I used the function FindRoot but I also have to choose a neighborhood for each parameter which is not obvious)
(Debug) In[]:=
NSolve[g==0,{γ1,δ1,γ2}]
(Debug) Out[]=
$Aborted
​