I received this plant from my daughter who recently purchased a home; the former owner left her all her houseplants including 3 pots of Madagascar Palm. This specimen is leaning precariously; too shallow of a pot and not heavy enough I believe. I plan to repot it but my question is can I plant it deeper so that the soil reaches the green and thicker portion of the stem. Note the narrow portion of stem that is above the soil at present. Can I bury that thin portion of the stem? This is not a type of plant I am familiar with so appreciate any advice.

Your plant looks like a Euphorbia (leuconeura I think?), which is technically not a Madagascar palm (Pachypodium lamerei) but is also from Madagascar. These plants tend to get a little leggy, especially indoors, and that may contribute to the instability you are observing, but they are kind of top heavy to start with.

You can put the plant into a bigger pot if you like, but try to find one that is wider than deep, otherwise underfill whatever you do have. A nice heavy ceramic pot ought to provide a good anchor for the bushy top growth. Do not overpot the plant; it's better to go up stepwise in small increments rather than make big leaps all at once. Make sure the new pot has a hole at the bottom. Use regular potting soil mixed with an equal volume of perlite/pumice/equivalent to make sure the soil has excellent drainage.

I would not bury the skinny part at the base of the stem. That is the hypocotyl (seed stem) of the plant and you may run the risk of rotting the plant if you bury it. I think a nice heavy pot will better serve your needs anyway.

The plant looks quite healthy, but be aware that it enjoys a lot of light indoors, basically as much light as you can provide. More light will help encourage a compact form in the years going forward and work to your favor in avoiding rot. Using soil with good drainage will also help you avoid rot. Water well when the soil has gone dry or almost dry at depth, but there is no advantage to allowing the soil to sit bone dry for any extended period of time.

In the winter it will probably have far fewer leaves and will probably enjoy less frequent water than now, which is basically prime growing season.

Thank you so much Baja; I really appreciate all your advice and correct id. I have the perfect sized ceramic pot available so that is good. I can provide southern exposure from large patio windows in the winter although our days in the north are quite short. Do you suggest putting it outdoors for the summer? Our summers are hot with long days and lots of sunshine and tend to be on the drier side; what kind of exposure would be best(after acclimating it of course)? I can provide sun, partsun or shade. Does it need to be protected from the rain and wind? Thanks again!

The southern exposure in winter is perfect. Again, expect some degree of rest during winter but the plant should not go totally leafless, especially with strong light. It would enjoy some outdoor sun in the coming months... warmth and light are both helpful. I don't think blazing full sun is the way to go, though. About half a day of direct sun probably would be best. Maybe 3-6 hours, in that range. An acclimation (over the period of weeks) will be helpful as you have noted.

This plant should enjoy roughly the same care as a crown of thorns plant (E. milii/hybrid) and can take a little extra water during the summer months. So summer rain is probably just fine unless it goes on for days and days on end. The wind may be an issue if you get strong gusts, because that leafy top part of the plant will act like a sail if the bottom part is not firmly anchored. I keep my Euphorbia in a position by a wall which blocks most of the prevailing winds.

Perfect!