  This paper discusses the construction of a new $(3+1)$-dimensional
Korteweg-de Vries (KdV) equation. By employing the KdV's recursion operator, we
extract two equations, and with elemental computation steps, the obtained
result is $
3u_{xyt}+3u_{xzt}-(u_{t}-6uu_{x}+u_{xxx})_{yz}-2\left(u_{x}\partial_{x}^{-1}u_{y}\right)_{xz}-2\left(u_{x}\partial_{x}^{-1}u_{z}\right)_{xy}=0
$. We then transform the new equation to a simpler one to avoid the appearance
of the integral in the equation. Thereafter, we apply the Lie symmetry
technique and gain a $7$-dimensional Lie algebra $L_7$ of point symmetries. The
one-dimensional optimal system of Lie subalgebras is then computed and used in
the reduction process to achieve seven exact solutions. These obtained
solutions are graphically illustrated as 3D and 2D plots that show different
propagations of solitary wave solutions such as breather, periodic, bell shape,
and others. Finally, the conserved vectors are computed by invoking Ibragimov's
method.
