\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-y \left ( t \right ) +\cases {x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}-1 \right ) \sin \left ( \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ^{-1} \right ) &$ \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}\neq 1$\cr 0&otherwise\cr },{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =x \left ( t \right ) +\cases {y \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}-1 \right ) \sin \left ( \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ^{-1} \right ) &$ \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2}\neq 1$\cr 0&otherwise\cr } \right \} } \]
Mathematica: cpu = 11.112911 (sec), leaf count = 106 \[ \text {DSolve}\left [\left \{x'(t)=\left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin \left (\frac {1}{x(t)^2+y(t)^2}\right ) x(t) \left (x(t)^2+y(t)^2-1\right ) & x(t)^2+y(t)^2\neq 1 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )-y(t),y'(t)=\left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin \left (\frac {1}{x(t)^2+y(t)^2}\right ) y(t) \left (x(t)^2+y(t)^2-1\right ) & x(t)^2+y(t)^2\neq 1 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \right )+x(t)\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 0 (sec), leaf count = 0 \[ \text {hanged} \]