  A new class of 2-orthogonal polynomials satisfying orthogonality conditions
with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak
K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence
relations and some properties. Integral Transforms Spec Funct.
2021;32(2):134-153]. Six interesting special cases were pointed out there. For
each case, we precisely deal with the integral representation problem for the
functionals associated to these polynomials. The focus is on the matrix
differential equation $\big({\bf\Phi U}\big)'+{\bf\Psi U}=0$, with ${\bf
U}={^t}(u_0 , u_1)$ and ${\bf\Phi}$, ${\bf\Psi}$ are $2\times2$ polynomial
matrices, from which we establish the differential equations satisfied by the
two functionals. Based on this, depending on the case, we show that $u_0$ and
$u_1$ are represented via weight functions supported on the real line or
positive real line and defined in terms of various special functions. In order
for certain integral representations to exist, addition of Dirac mass is
necessary.
