We find that the slope of the line is $-\frac{3}{5}$. Finding the equation of the line by plugging in points yields $3x+5y=43$. Observation tells us that $y$ is an integer when $3x+12$ is a multiple of $5$, or when $x+4$ is a multiple of $5$. These occur every $5$ units apart, so the values are $-4,1,6,11,16$. We can plug in to find the $y$ values. Thus all the points are $(-4,11),(1,8),(6,5),(11,2),(16,-1)$.