Clear[p, Uinit, ff1, ff2, FluxFunc, A, FlowLine1, FlowLine2, 
      v, θ, θ0, θ1, θ2, ϕ, u, xnaught, x, 
      icu,  icv, t, Vinitial]
<< Graphics`PlotField`

p[v_] = 2  v^(-2)
Uinit = {2, 1}
α =  .5
β =  .5

FluxFunc[u_, 
        v_] = {ff1[u_, v_],  
          ff2[u_, v_]} = {p[v], -u}
A = {{(∂_u) ff1[u, v ], (∂_v) 
            ff1[u, v]}, {(∂_u) ff2[u, v], (∂_v) 
            ff2[u, v]}} 
MatrixForm[A]

{EVal1[v_], Eval2[v_]} = 
      Eigenvalues[A]

{EVec1[v_], EVec2[v_]} = 
      Eigenvectors[A]

IntegralCurve1[θ_] = (∫_0.01^θ) 
              EVec1[θ] \[DifferentialD]θ + 
        Uinit
IntegralCurve2[θ1_, θ2_] = (∫_0.01^θ)2 
              EVec2[θ] \[DifferentialD]θ + 
        IntegralCurve1[θ1]

PlotVectorField[
      FluxFunc[u, v], {u, 0, 5}, {v, 0, 5}]
Show[
      ParametricPlot[
        IntegralCurve1[θ], {θ,  .01, α}], 
ParametricPlot[
        IntegralCurve2[α, θ], {θ,  .01, β}]]