  We consider the case of exceptional Laguerre polynomials $X_1$ of type I, II
and III, their ordinary differential equations and the problem of finding
general solution beside the polynomial part. We will develop an algebraic
approach based on the Schrodinger form of the problem and associate
representations of the underlying spectrum generating algebra. We use the
Darboux-Crum transformation to construct ladder operators of fourth order for
the case of the exceptional Laguerre polynomials $X_1$ of type I, II and III.
We then obtain all zero modes for the lowering and raising operators. We
construct the induced representation for the linearly independent solutions,
including the polynomial states. Those states forming the general solution are
important non only in the construction of wider set of physical states
satisfying different boundary conditions, but also in context of getting
isospectral deformations. Our approach, allows to provide a completely
algebraic construction of the two linearly independent solutions of the
ordinary differential equation of the exceptional orthogonal polynomials of
Laguerre type $X_1$ ( case I, II and III). The analogue of Rodrigues formulas
for the general solution are constructed. The set of finite states from which
the other states can be obtained algebraically is not unique but the vanishing
arrow and diagonal arrow from the diagram of the 2-chain representations can be
used to obtain minimal sets. These Rodrigues formulas are then exploited, not
only to construct all the states (polynomial and non-polynomial), in a purely
algebraic way, but also to obtain coefficients from the action of the ladder
operators also in an algebraic manner. Those results are established via higher
commutation relations related to the cubic Heisenberg Weyl algebra.
