If $y\in I$ then $y\in \left\{-7,-6,....,\mathrm{0,1},...,\mathrm{6,7}\right\}$
Total integral points on ${\left(\frac{x}{2}\right)}^2+y^2=50$ are $12$ which are
$\left\{\begin{array}{l}\left(\mathrm{2,7}\right),\left(\mathrm{14,1}\right),\left(2,-7\right)\left(-\mathrm{14,1}\right),\left(-\mathrm{2,7}\right),\left(14,-1\right)\\ \left(-2,-7\right),\left(-14,-1\right),\left(\mathrm{10,5}\right)\left(10,-5\right),\left(-\mathrm{10,5}\right)\left(-10,-5\right)\end{array}\right\}$
Total number of lines passing through at least two points among these are ${\mathrm{}}^{12}{C}_2=66$ in which $x=2,x=-2, x=14, x=-14, x=10, x=-10,$ $y=1, y=7, y=-1, y=-7, y=5, y=-5$ are rejected.
$\Rightarrow$ Total number of lines are $66-12=54.$