We consider the Lotka-Volterra Equations
$$\dot{x}=x(1+ax+by),\qquad \dot{y}=y(-\lam+cx+dy),$$ with $\lam$
a non negative number. Our aim is to understand the mechanisms
which lead to the origin being linearizable, integrable or
normalizable. In the case of integrability and linearizability,
there is a natural dichotomy. When the system has an invariant
line other than the axes, then the system is integrable and we
give necessary and sufficient conditions for linearizability in
this case. When there is no such line, then the conditions for
linearizability and integrability are the same. In this case we
show that the monodromy groups of the separatrices play a key
role. In particular for $\lam = p/q$ with $p+q\le12$ and $\lam=
n/2, 2/n$ with $n\in{\bf N}$ the origin is linearizable if and
only if the monodromy groups can be shown to be linearizable by
elementary arguments. We give 4 classes of these conditions, and
their duals, in terms of the parameters of the system, and
conjecture that these, together with two exceptional cases of
Darboux linearizability, are the only integrability mechanisms for
rational values of $\lam$.
The work on normalizability is more tentative. We give some
sufficient conditions for this via monodromy groups, and give a
complete classification when $\lam = 0$. We also investigate in
detail the case $\lam=1$, with $a+c=0$. Much of our ideas here
are based on recent work on the unfolding of the Ecalle-Voronin
modulus of analytic classification \cite{MRR}. In particular we
give examples of \lq\lq half-normalizable" systems as well as an
experimental example of a \lq\lq transcritical bifurcation" of the
functional moduli associated to the critical point.