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Vector field with ImplicitPlot ?

I wish to do plot a vector field plot of an implicit solution.

Equations :

(*V[x] is a polynomial function defined earlier *)
e[x_, v_] := 1/2 v^2 + V[x];
t3 = Table[e[x, v] == e1, {e1, 0, 1, 0.2}];
ImplicitPlot[Evaluate[t3], {x, \(-2\), 2}];

This plots a nice implicit plot for various values of e1. What I wish to do is to attach a 
"sense" to the contours (an arrow that describes the direction of the orbit).

However, I am at a loss as to how to do it with PlotVectorField.

Any help would be appreciated.

PS : Is there a way in which I can "automate" the labelling of the curves with the values 
of e1 they are plotted for (something better than manually editing the labels to the plot 
each time I change the range of e1) ?

3:29pm up 2:28, 1 user, load average: 0.07, 0.20, 0.16

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