Given : a^{2} = bc + 1 ---- ( 1 )

And

b^{2} = ca + 1 ---- ( 2 )

Now we subtract equation 2 from equation 1 and get

a^{2} - b^{2} = bc - ca

( a + b ) ( a - b ) = c ( b - a )

( a + b ) ( a - c ) = - c ( a - b )

( a + b ) = - c

c = - ( a + b ) , Substitute that value in equation 1 we get

a^{2 }= b [ - ( a + b ) ] + 1

a^{2} = - ab - b^{2} + 1

a^{2} + ab + b^{2} - 1 = 0

So, the integers are ,

a = 0 , b = - 1 and c = 1

a = 0 , b = 1 and c = - 1

a = 1 , b = - 1 and c = 0

a = 1 , b = 0 and c = - 1

a = - 1 , b = 0 and c = 1

a = - 1 , b = 1 and c = 0