Fine lines around the eyes can be related to the muscle squeezing and creating wrinkles, or it can result from actual changes in the skin. Small amounts of Botox can sometimes be used, and retinol skin products can be helpful in creating a skin tightening.
Without a photo, only general remarks can be made. However, if lines or "donut" rolls are present, a small droplet of Botox (or Dysport or Xeomin) placed in the lower lid a couple of millimeters below the lashes and along the midpupillary linel may help to improve them. If bags and tear troughs are a problem, Belotero Balance, a hyaluronic acid filler that possesses little tendency to cause the unwanted bluish Tyndall Effect when injected superficially might make an excellent choice. Consultation with a board certified aesthetic core physician with extensive experience and expertise in treating the periorbital area is strongly advised and be sure to ask to see befores and afters photo of his/her work.
Lines under the eyes can be cause from several things and their treatment will be different based on the cause. I would recommend and examination/consultation with a Dermatologist to discuss your treatment options.
WIthout an exam, it is difficult to say what could be done for what you describe as under eye lines. Some may be due to animation ( similing).
With muscular contraction, you will always see some lower lid lines, but there are ways to minimize them, sometimes botox, or a peel or ThermiSmooth may decrease the lines, but you need to be seen to know what can be done.
You may do well with a small amount of Botox to the lower eyelid muscle that is usually responsible for folding and wrinkling that skin in young people. If you have a prominent tear trough groove, then filler may also be helpful. It is best if you can provide a photo.