Can this system of equations be solved in real numbers?

[anemone]

Solve in real numbers the system below:

$a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$

 

[Wilmer]

Spoiler

a=1, b=1, c=1
a=-1, b=-1, c=-1

 

[MarkFL]

Spoiler

a=1, b=1, c=1
a=-1, b=-1, c=-1

Would you mind sharing how you found the solution? :D

 

[Wilmer]

Lazily, by inspection:
ab + ac - a^4 = 1
ab + bc - b^4 = 1
ac + bc - c^4 = 1

 

[anemone]

Spoiler

a=1, b=1, c=1
a=-1, b=-1, c=-1

Hi Wilmer,

Spoiler

Your answer (without the working, hehehe...) is correct, but the question remains on how we are going to prove those are the only solutions.(Nod)