I have a system of n x 1 equations

0 = A vec(xx') + B x + C

where

x is a n x 1 vector of unknowns

x' means transpose

vec means xx' has been vectorized so has dimension n^2 x 1

A is a known matrix with dimensions n x n^2

B is a known matrix with dimensions n x n

C is a known vector of dimension n x 1

I can solve these problems using a nonlinear solver. However, I am trying to find out if there are any theoretical results on how to solve this class of problems. I can find a lot of work on solving matrix quadratic equations. However, I can't find anything that has specifically this form, and I cannot figure out if I can rewrite this system in a way that is equivalent to other matrix quadratic problems I come across.