\documentclass[11pt]{scrartcl}
\usepackage{tikz}
\usetikzlibrary{3d}
\begin{document}
\begin{tikzpicture}
[x={(-0.2cm,-0.4cm)}, y={(1cm,0cm)}, z={(0cm,1cm)}, scale=2,fill
opacity=0.80,color={gray}]
\tikzset{zxplane/.style={canvas is zx plane at y=#1,very thin}}
\tikzset{yxplane/.style={canvas is yx plane at z=#1,very thin}}
\begin{scope}[yxplane=-1]
\path[shade,draw] (0,0) circle[x radius = 2 cm,y radius =1 cm] ;
\path[shade,draw] (1,0) circle[x radius = 0.6 cm,y radius =0.3cm] ;
\path[shade,draw] (-1,0) circle[x radius = 0.6 cm,y radius =0.3cm] ;
\path coordinate (a1) at (-1,0)
coordinate (a2) at (-1.2,0)
coordinate (b1) at (1.2,0);
\end{scope}
\begin{scope}[yxplane=1]
\path[shade,draw] (0,0) circle[x radius = 2 cm,y radius =1 cm] ;
\path[shade,draw] (1,0) circle[x radius = 0.6 cm,y radius =0.3cm] ;
\path coordinate (a) at (-1,0)
coordinate (b) at (1,0);
\end{scope}
\draw (a)--(a1) (a)--(a2) (b)--(b1);
\foreach \point in {a,a1,a2,b,b1}
\fill [black,opacity=.5] (\point) circle (1pt);
\end{tikzpicture}
\end{document}