DSolve[{Derivative[1][x][t] == -(y[t]*(x[t]^2 + y[t]^2)), Derivative[1][y][t] == Piecewise[{{x[t]^2 + y[t]^2, x[t]^2 + y[t]^2 >= 2*x[t]}}, (x[t]^2 + y[t]^2)*(x[t]/2 - y[t]^2/(2*x[t]))]},{x[t], y[t]},t]