Multiresponse regression with some eigenvector constraints

Hey I'm dealing with a regression problem where I have a response matrix Y, data matrix X and some parameter matrix B. Let's say that their sizes are (N,m), (N,p) and (p,m) respectively.

To give you some insight then Y are actually histogram observations (approximations of a distribution) which I'm trying to predict hence there is a normalization constraints. I also want to use a linear model and find optimal B with respect to some loss e.g. the Frobenius norm.

The constraints are the ones that I'm puzzled by and don't have an exact idea on how to deal with. They can be formulated in such a way that:

XB1 = 1 i.e. unitary vector (1) is an eigenvector of the matrix (X*B) with an eigenvalue 1.

Is there another way to formulate these constraints? Can you use SVD to solve such an optimization problem?

If you could at least point me in the right direction, cause I'm not exactly sure what to put in the search bars, then that would be awesome :)

edit 1 : Another way to formulate the constraint: (XB - Y)1 = 0

Edit 2: sorry it's not called unitary, it's just a vector with all entries as 1s.