How would one go about solving the system of five equations:

${p}^{2}=p+q-2r+2s+t-8$

${q}^{2}=-p-2q-r+2s+2t-6$

${r}^{2}=3p+2q+r+2s+2t-31$

${s}^{2}=2p+q+r+2s+2t-2$

${t}^{2}=p+2q+3r+2s+t-8$

over the integers?

${p}^{2}=p+q-2r+2s+t-8$

${q}^{2}=-p-2q-r+2s+2t-6$

${r}^{2}=3p+2q+r+2s+2t-31$

${s}^{2}=2p+q+r+2s+2t-2$

${t}^{2}=p+2q+3r+2s+t-8$

over the integers?