$HPT \Leftrightarrow \left\{ \begin{array}{l} x+y=1\\ x^3+y^3-x^2-y^2=0 \end{array} \right.$$\Leftrightarrow \left\{ \begin{array}{l} x+y=1\\ x^2(x-1)+y^2(y-1)=0 \end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l} x+y=1\\ x^2y+xy^2=0 \end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l} x+y=1\\ xy(x+y)=0 \end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l} x+y=1\\ xy=0 \end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l} x=0\\ y=1 \end{array} \right.$ or $\left\{ \begin{array}{l} x=1\\ y=0 \end{array} \right.$