lie down under (something)
To rest or recline on one's back beneath something. We lay down under the large oak tree in the fields behind grandfather's cottage to escape the heat of the sun. All I want to do on vacation is lie down under an umbrella on the beach and sleep or read.
lie down under something
to lie down beneath something. She was tired, so she lay down under a willow tree by the brook. The dog lay down under a lawn chair and slept.