Answer: Based on your description I suspect the plant is infested with scale, a fairly common pest on this plant. The insect damage causes sap to drip.
Here is a photo: http://www.ifas.ufl.edu/~insect/orn/scales/tessellated_scale...
Scale can be removed gently by hand (scrape with your fingernail, very tedious and difficult to get them all) but I would suggest you contact your county extension for their most up to date recommendation for the most appropriate control of this pest on this particular plant kept in the home. Also, if the problem is not scale, then they should be able to identify it for you and tell you what to do about it.
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