tl;dr: To the narrow question:
Is a solar free-return trajectory possible?
Yes, a trajectory that starts near the Earth, goes around the Sun, and returns to a place near Earth is possible.
I'm not an expert but let me try to clear up some points though:
- Traditionally, a free return trajectory has been envisioned starting from the large close-to-fixed body (e.g. Earth), swinging past a small body that's orbiting around it (e.g. Moon), then returning to the large body.
- As @Hobbes points out you need a big impulse to go from low orbit around the large body to the free-return orbit.
- A free-return orbit is not necessarily the same thing as a cycler orbit; if you didn't put on the breaks after the first complete cycle and swung back out a second time, the small body would not be there the next time in most practical cases, and then you'd be in a pickle.
I'm coining a new phrase: "Dyson circuit".
So I think you are looking for periodic solutions, and there are certainly quasi-periodic 3-body orbits that can pass alternately near the Sun and near the Earth. In reality perturbations would not allow these to be quasi-periodic for very long, but at least they could be station-kept.
From this answer to the question What sort of orbital elements are used to describe halo orbits?:
The extremely cool and colorful paper E. J. Doedel et al, (2007) Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem International Journal of Bifurcation and Chaos 17, 2625 (2007). https://doi.org/10.1142/S0218127407018671 builds a system of illustrations that show all of the known, periodic, orbits in the CR3BP (Circular Restricted Three-Body Problem). This includes many kinds or classes of orbits but excludes Lissajous Orbits because they are not in general periodic. (note: ignore the drawing in the Wikipedia article!)
You can and probably should also download the paper from its non-paywalled ResearchGate site, make some coffee, then spend six months enjoying it.
Here are some 3-body orbits that might do the job you are looking for. As mentioned, in the real world you would have to include some station-keeping to maintain the orbit over time. The images are in a rotating frame.
These images are drawn using the Earth-Moon system, but you can also imagine the Sun-Earth system where the Moon in the picture is the Earth, and the Earth in the picture is the Sun.
These are schematically drawn, when you change to the much larger mass ratio of the Sun-Earth system, the shapes will change.
It's not a coincidence that these two resemble the two cis-lunar co-planar free-return orbits in the last plot below!
Back to 1963!
Below are four free-return trajectories starting from Earth orbit and swinging past the Moon. These are drawn in a rotating frame of reference so that both the Earth and Moon are pinned to the x-axis of these plots. These are from NASA Technical Note D-1833; Trajectories in the Earth-Moon Space with Symmetrical Free Return Properties written in 1963 by Arthur J. Schwaniger at the George C. Marshall Space Flight Center.
In the paper, a free-return is defined:
For manned exploration the first flights will very likely be such "fly-by" trips with no plan for landing on the moon. Indeed, when the manned mission is to land on the moon's surface, a "free return" trajectory may be used so that if unforeseen difficulties arise which would make landing undesirable or impossible (particularly a failure of the propulsion system to brake the vehicle speed so as to make landing possible) the astronauts will return safely to the earth.
They are all free-return trajectories starting and ending near the Earth, and except in special cases they work one-time only. You need a big propulsive maneuver to enter into one from LEO in the beginning, and you need a 2nd big propulsive maneuver to get back into LEO (or reenter the atmosphere) when you get back to Earth.
They're called symmetrical because the 2nd half is nearly a mirror-image of the first half.
There are certain combinations of injection altitudes and velocities that can produce periodic or cycler-like behavior but in general these will not approach the Moon very closely.
In a non-rotating frame I think that the