The equation $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.

I’m looking for positive integer solutions to $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.

Background.

I was looking at “Primes which are the sum of three nonzero 8th powers” https://oeis.org/A283019 and the like, and wondered if there are rules similar to those for a prime being the sum of two squares.

My efforts.
I’ve found primes of the form $x^n+y^n+z^n$ up to $n=19$.

For $n=1$ solutions are trivial, for $n=2$ they are abundant. There are plenty of results for $n=3$ and they’re not uncommon for $n=4$.

However, for $\color{blue}{n=5}$ I’ve found just two lonely solutions.

$$(n,x,y,z=u,v,w,p)$$

$$(2,3,4,4=1,2,6,41)$$
$$(2,3,5,5=1,3,7,59)$$
$$(2,3,5,7=1,1,9,83)$$

$$(3,1,5,5=2,3,6,251)$$
$$(3,4,6,9=1,2,10,1009)$$
$$(3,1,9,9=4,4,11,1459)$$

$$(4,9,16,16=8,13,18,137633)$$
$$(4,4,18,19=1,6,22,235553)$$
$$(4,8,16,21=6,13,22,264113)$$

$$(\color{blue}5,11,183,209=19,168,216,604015282243)$$
$$(\color{blue}5,481,782,788=321,772,808,622015202536001)$$

My question.
Can anyone find more solutions for $n=5$, or $n>5$, or give any insights, please.

Solutions Collecting From Web of "The equation $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime."

Taxicab numbers can be generalised to any number of terms.

With credit to Duncan Moore, author of Generalised Taxicab Numbers and Cabtaxi Numbers where I found suitable solutions, I can contribute:
$$809^6+1851^6+2443^6=1277^6+1491^6+2489^6=253089021060516507491$$
$$511^6+2945^6+3285^6=1339^6+2457^6+3449^6=1909058509267895080811$$

A web search throws up this PDF entitled Complexity of Finding Values of the Generalized Taxicab Number which concludes that the problem is “at least supposed to be NP-Hard”.

Using Duncan Moore’s solutions, I’ve found that solutions for $n=5$ are not as rare as I thought earlier.
Here are a few of the smallest; many of these also have $x+y+z=u+v+w$.

$$80^5+219^5+270^5=132^5+154^5+283^5=1941923897099$$
$$317^5+1008^5+1052^5=413^5+642^5+1172^5=2332329237198157$$
$$195^5+537^5+1267^5=970^5+1007^5+1072^5=3309936251919439$$
$$125^5+395^5+1479^5=559^5+922^5+1448^5=7086510650848399$$
$$893^5+1039^5+1691^5=733^5+1231^5+1659^5=15605379807526343$$
$$519^5+911^5+1841^5=103^5+1431^5+1737^5=21813107171029351$$
$$167^5+1153^5+1843^5=483^5+809^5+1871^5=23300964032544043$$
$$58^5+1171^5+1920^5=198^5+1331^5+1890^5=28293760481131619$$
$$101^5+1092^5+1944^5=274^5+906^5+1957^5=29316755331623957$$
$$69^5+1559^5+1863^5=679^5+823^5+1989^5=31651525808548691$$
$$235^5+1669^5+1965^5=189^5+1817^5+1863^5=42247371948274349$$
$$1124^5+1798^5+2119^5=1223^5+1644^5+2174^5=63307290246821191$$
$$283^5+2042^5+2182^5=432^5+1744^5+2331^5=84968164403859307$$
$$491^5+1459^5+2407^5=332^5+1653^5+2372^5=87433928094323557$$
$$58^5+1828^5+2367^5=981^5+1259^5+2463^5=94712207910987743$$
$$1266^5+1678^5+2469^5=1117^5+1873^5+2423^5=108305315229854293$$
$$693^5+1693^5+2497^5=417^5+2213^5+2253^5=111140193938890643$$
$$997^5+1979^5+2641^5=1009^5+1962^5+2646^5=159821917701479857$$
$$877^5+1960^5+2666^5=770^5+2252^5+2541^5=164123524321248733$$
$$1438^5+1916^5+2777^5=1132^5+2478^5+2521^5=197120435154165401$$
$$1277^5+2208^5+2784^5=1557^5+1849^5+2863^5=223118389276174349$$
$$942^5+2461^5+2808^5=413^5+1407^5+3041^5=265591178424588301$$
$$1912^5+1946^5+3083^5=804^5+2646^5+2891^5=331987128193291451$$
$$969^5+2791^5+2881^5=1341^5+2129^5+3171^5=368689558572449201$$
$$1799^5+2801^5+3543^5=2143^5+2371^5+3629^5=749539223719548943$$
$$559^5+2869^5+3671^5=1231^5+2039^5+3829^5=861122005375239499$$