\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-y \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =\cases { \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}&$2\,x \left ( t \right ) \leq \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}$\cr \left ( 1/2\,x \left ( t \right ) -1/2\,{\frac { \left ( y \left ( t \right ) \right ) ^{2}}{x \left ( t \right ) }} \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) &otherwise\cr } \right \} } \]
Mathematica: cpu = 2.354299 (sec), leaf count = 86 \[ \text {DSolve}\left [\left \{x'(t)=-y(t) \left (x(t)^2+y(t)^2\right ),y'(t)=\left ( \begin {array}{cc} \{ & \begin {array}{cc} x(t)^2+y(t)^2 & x(t)^2+y(t)^2\geq 2 x(t) \\ \left (x(t)^2+y(t)^2\right ) \left (\frac {x(t)}{2}-\frac {y(t)^2}{2 x(t)}\right ) & \text {True} \\ \end {array} \\ \end {array} \right )\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 0 (sec), leaf count = 0 \[ \text {hanged} \]