Graphics, Figures & Tables ⇒ Plot using tikz

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\documentclass[tikz,border=13mm]{standalone}
\usetikzlibrary{arrows.meta,decorations.markings}
\usetikzlibrary{calc}
%\pgfplotsset{ticks=none}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}[>=stealth,declare function={f(\x)=\x^3/3-\x;}]
 \begin{axis}[name=MyAxis,grid=major,grid style=dotted , xmin=-2.2, xmax=2.2, ymin=-1.2, ymax=1.2,
          xlabel=$x\rightarrow$, ylabel={$y \rightarrow$}, 
				  xtick = {-2,-1,0,1,2},  xticklabels = {-2,-1,0,1,2},
					ytick = {-1,-.5,0,.5,1}, yticklabels={-1,-.5,0,.5,1},  %yticklabels=\empty,
         scale=1, restrict y to domain=0:.4,clip=false]
  \draw[scale=1,domain=-2.15:2.15,smooth,variable=\x,blue,thick] plot ({\x},{\x^3/3-\x});
  \draw[scale=1,red,fill=red] (1, -2/3) circle (.5ex);
  \draw[scale=1,red,fill=red] (-1, 2/3) circle (.5ex);
  \draw[scale=1,black] (0, 0) circle (.5ex);
    \draw[scale=1,red,thick,>={Stealth[round]},postaction=decorate,
  decoration={markings,
  	mark=at position 0.15 with {\arrow[black]{<.<}},
  	mark=at position 0.4 with {\arrow[black]{<}},
  	mark=at position 0.65 with {\arrow[black]{<.<}},
  	mark=at position 0.925 with {\arrow[black]{<}},
  }] % (-2,{f(-2)}) --(0,{f(.96)}) --  plot[variable=\x,domain=.96:.3479]  (\x,{f(\x)}) -- cycle; % Gamma_0 curve
(-2,{f(-2)}) -- (1,{f(1)})-- plot[variable=\x,domain=1:2] (\x,{f(\x)})--(-1,{f(-1)})--plot[variable=\x,domain=-1:-2] (\x,{f(\x)});
				\draw[scale=1,black,thick,rounded corners=8pt,postaction=decorate,
		decoration={markings,
			mark=at position 0.15 with {\arrow[black]{<.<}},
		mark=at position 0.4 with {\arrow[black]{<}},
		mark=at position 0.65 with {\arrow[black]{<.<}},
		mark=at position 0.925 with {\arrow[black]{<}},
		}] (-2,{f(-2)-0.05}) to[bend right=5] (1.05,{f(.95)}) --  plot[variable=\x,domain=1.05:2.05] (\x,{f(\x)+0.05}) --(-1-0.05,{f(-1)+0.05})--plot[variable=\x,domain=-1-0.05:-2+0.05] (\x,{f(\x)-0.05})--cycle;
				\end{axis}
\end{tikzpicture}
\end{document}

I want to get a plot like attached 1234.png file. I got sa.jpg. My code works perfect except at two corners of the black curve near the solid circles. Please correct my code.