The charm of elliptic curves is that given one or two integer points, one can find others by the group law. However the easy to guess points from the title just pump me around trough a cyclic group of order six, so they are no help for the question whether there are any others.
My interest in this curve comes from the question if there are any triangular numbers that are also cubes, beyond the number 1. So we want to solve $\frac{m(m-1)}{2} = n^3$ and the substitution $y = 2m -1, \hspace{.1cm} x = 2n$ gives the above curve. This means that I am mostly interested in integer points $(x, y)$ for which $x$ is even and $y$ is odd, while both are positive; as expected there is only one of these among the easy to guess points (corresponding to $\frac{m(m-1)}{2} = n^3 = 1)$ but are there any others?
Those are the only integer points; in fact, they are the only rational points. For a lovely descent proof due to Euler, see Conrad’s paper.