= Solution
Cubing and adding the parametrized coordinates yields
$$
x^3+y^3=\frac{27a^3t^3(1+t^3)}{(1+t^3)^3}
=\frac{27a^3t^3}{(1+t^3)^2}=3axy.
$$
Therefore the implicit <algebraic curve> is
$$
\boxed{x^3+y^3-3axy=0}.
$$
For $a\ne0$, every nonorigin real point on this curve has $x\ne0$ and can be recovered using $t=y/x$; the origin is already included. The equation is invariant under exchanging $x$ and $y$, so its reflection line is $\boxed{y=x}$. In the parametrization that reflection corresponds to replacing $t$ by $1/t$. If $a=0$, the implicit cubic degenerates to the line $x+y=0$, whereas the given parametrization only describes the origin, so this degenerate case must be kept separate.
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