Pythagorean type diophantine equation. 
How to find all solutions to
$$ a^2+b^2+c^2+d^2=e^2+2$$
where all variables $a$ to $e$ are positive integers and $e^2 \equiv 1 \mod 8$

I tried using parameterization similar to pythagoras equation, but no success so far. Any help will be appreciated.
Thanks!

For such equations, you can use the standard approach. 
One approach is to use equations Pell.  For the beginning will talk about a more simple way.
I. Eq.1 
$$a^2+b^2+c^2+d^2=(2q+1)^2+2$$
Solutions have the form:
$$a=x + k$$
$$b=x + p$$
$$c=x + s$$
$$d=x - z+1$$
$$q=x$$
where,
$$x = -1 -z+ (z^2 - k p - k s - p s),\quad z = k+p+s$$
for any $k,p,s$. 
II. Eq.2 
$$a^2+b^2+c^2+d^2=(4q+1)^2+2$$
Solutions have the form:
$$a=2(y + k)$$
$$b=2(y + p)+1$$
$$c=2(y + s)+1$$
$$d=2(y - z)-1$$
$$q=y$$
where,
$$y = -k + 2z + 2(z^2 - k p - k s - p s),\quad z = k+p+s$$ 
for any $k,p,s$.
III. Eq.3 
$$a^2+b^2+c^2+d^2=(8q+1)^2+2$$
Solutions have the form:
$$a=16k^2+4p^2+4s^2+8pk+8ks+4ps+2s+2p-1$$
$$b=16k^2+4p^2+4s^2+8pk+8ks+4ps+4s+4p+8k$$
$$c=16k^2+4p^2+4s^2+8pk+8ks+4ps+4s+6p+4k+1$$
$$d=16k^2+4p^2+4s^2+8pk+8ks+4ps+6s+4p+4k+1$$
$$q=4k^2+p^2+s^2+2pk+2sk+ps+s+p+k$$
$k,p,s$ - integers asked us.
