\(\dot{x}_1x_1 = -\dot{x}_2x_2\)
\(\int x_1 \frac{dx_1}{dt} = -\int x_2 \frac{dx_2}{dt}\)
\(\frac{x^2_1}{2} + c_a = -\frac{x^2_2}{2} + c_b\)
\(\frac{x^2_1}{2} + \frac{x^2_2}{2} = c_b - c_a\)                    With \(K=2(c_b - c_a)\)
\(x^2_1 + x^2_2 = K\)