For a fixed generalized reflection matrix $P$, i.e., $P^T=P, P^2=
I$, and $P\neq \pm I$, then a matrix $A $ is called a symmetric
$P$-symmetric matrix if $A=A^T$ and $(PA)^T=PA$. This paper is
mainly concerned with finding the least squares symmetric
$P$-symmetric solutions to the matrix inverse problem $AX=B$ with a
submatrix constraint, where $X$ and $B$ are given matrices of
suitable size. By applying the generalized singular value
decomposition and the canonical correlation decomposition, an
analytical expression of the least squares solutions is derived
basing on the Projection Theorem in Hilbert inner products spaces.
Moreover, in the corresponding solution set, the analytical
expression of the unique minimum-norm solution is described in
detail.