Find all integers $a$,$b$ and $c$ that satisfy the following equations

$a^2=bc+1\tag{i}$

$b^2=ac+1\tag{ii}$

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I tried solving came out with following results:

$(i)-(ii)$
gives
$a+b+c=0$
which implies,
$a^3+b^3+c^3=3abc$

Multiplying equation $(i)$ with $a$ and multiplying equation $(ii)$ with $b$
and then subtracting $(i)-(ii)$
we get 
$a^2+b^2+ab=1$
which can be written as 
$(a+b)^2-ab=1$ , which can be further written as $c^2-ab=1$.