<< Graphics`PlotField`

a = 
      PlotVectorField[{x - y, 1 - x^2}, {x, -3, 3}, {y, -3, 3},  
        Axes -> True]

soln = 
      NDSolve[{x'[t] == x[t] - y[t], y'[t] == 1 - x[t]^2, 
          x[0] == 1/2, y[0] ==  .5}, {x, y}, {t, -3, 
          4}]

p1 = 
      ParametricPlot[
        Evaluate[{x[t], y[t]} /. soln], {t, -3, 4}]

Show[a, p1, PlotRange -> {{-4, 4}, {-4, 4}}]