Envelope of Ellipses with Common Major-axis Length are the envelopes of the following set of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem;  
$$\mathcal{E}:=\Biggr\{\frac{(x-e)^2}{a^2} +\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a)$$
the foci of these ellipses have coordinates $(0,0)$ and $(2e,0)$
 A: For the current version of the question, we can just derive the equation of the envelope in  the usual manner. Using Mathematica to do the tedious algebra on my behalf, we obtain:
First[Factor[GroebnerBasis[{(x - e)^2/a^2 + y^2/(a^2 - e^2) == 1,
                            D[(x - e)^2/a^2 + y^2/(a^2 - e^2) - 1, e] == 0}, {x, y, a}, e]]]
   (4 a^2 - x^2 - 4 a y) (4 a^2 - x^2 + 4 a y) (x^2 + y^2)

where we see the envelope has three components: an isolated point at $(0,0)$ and two parabolas (which are certainly "classical curves of mathematics" as the OP asks), $y=\pm\dfrac{x^2}{4a}\mp a$:
With[{a = 1}, 
     Show[ParametricPlot[Table[{e + a Cos[t], Sqrt[a^2 - e^2] Sin[t]}, {e, 0, a, a/24}],
                         {t, 0, 2 π}], 
          Plot[{x^2/(4 a) - a, a - x^2/(4 a)}, {x, -2, 2}, 
               PlotStyle -> Directive[AbsoluteThickness[4], ColorData[97, 3]]], 
          AspectRatio -> Automatic, PlotRange -> {{-2, 2}, Automatic}]]


A: What do you mean by the "envelope" of the $\mathcal{E}_1$ set of ellipses?

          


          

$\mathcal{E}_1$: $a=1$; $e=0,0.1,\ldots,0.9,1$.


Added after OP change, now $\mathcal{E}$:

          


          

$a=2$; $e=0,0.1,\ldots,a$.


