This is the wrong forum for your question.
The short answer is yes, there is. I assume x,y, and z are real numbers. For complex numbers or other rings, you will need results from another field of study, possible algebraic geometry.
The first two equations represent a plane possibly intersecting a sphere. There are three cases:
no intersection. Thus no solution. Stop
intersection is a point. Then x=y=z and thus c had better be 3(a/3)^8, otherwise no solution. (b and a have to satisfy a relation for this case to occur.)
intersection is a circle. Then use geometry or Lagrange multipliers to find the image of this circle under the map $x^8 + y^8 + z^8$. Since the circle is compact and the map is continuous, the range is a closed interval. If c is in this range there are anywhere from 1 to at least 8 distinct solutions. If c is not in the interval, no solutions.
Regarding the question of publishability (mentioned in a comment), this result would not be considered appropriate for most of not for all research journals. Even if there is novelty in the approach, one would expect the approach to be much more widely applicable to be considered. If it were presented in a pedagogically engaging way, it might serve as an example lesson for undergraduate students. How one would present this solution is a question for another forum.
Gerhard "You Could Blog This Elsewhere" Paseman. 2019.03.13.