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--- abstract: 'The form factors of the descendant operators in the massive Lee-Yang model are determined up to level 7. This is first done by exploiting the conserved quantities of the integrable theory to generate the solutions for the descendants starting from the lowest non-trivial solutions in each operator family. We then show that the operator space generated in this way, which is isomorphic to the conformal one, coincides, level by level, with that implied by the $S$-matrix through the form factor bootstrap. The solutions we determine satisfy asymptotic conditions carrying the information about the level that we conjecture to hold for all the operators of the model.' --- SISSA 09/2005/FM \ \ \ [*via Beirut 2-4, 34014 Trieste, Italy*]{}\ [*INFN sezione di Trieste*]{}\ [*E-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>*]{}\ Introduction ============ A massive quantum field theory can be seen as the perturbation of a fixed point (conformal) theory by relevant or marginally relevant operators. The perturbation spoils some simple features of the conformal point, as the power law behavior of two-point functions, but preserves some structural properties. In particular, the operator space of the massive theory is expected to be isomorphic to that of the conformal theory. In principle, this expectation can be checked in $1+1$ dimensions even for non-trivial fixed points, due to the existence of integrable quantum field theories. What makes the check far from obvious is that the operator content at and away from criticality is determined in two completely different ways. On one hand, the operator content at the conformal point is dictated by the representation theory of the Virasoro algebra underlying the infinite dimensional conformal symmetry in two dimensions [@BPZ]. This reveals a structure of operator families, each consisting of a primary operator and an infinite tower of descendants organized into multiplets (levels) labeled by integer spaced scaling dimensions. On the massive side, instead, integrable quantum field theories are solved through the exact determination of the $S$-matrix [@Taniguchi]. The operators are then constructed determining their matrix elements (form factors) on the particle states as solutions of a set of functional equations [@KW; @Smirnov]. It was first shown in [@CM] for the thermal Ising model, and later for more complicated models [@Koubek; @Smirnovcounting; @JMT], that the global counting of independent solutions of the form factor equations matches that expected from conformal field theory. This counting procedure for the form factor solutions, however, disentangles neither the operator families sharing the same symmetry properties nor the levels of the descendants. In this paper we tackle the problem of the detailed isomorphism between the critical and off-critical operator spaces for the simplest interacting massive theory, the Lee-Yang model. The previously available results for the operators of this model concern the non-trivial primary operator [@Alyosha] and the first non-trivial scalar descendant $T\bar{T}$ of the identity family [@ttbar]. We exploit the first few quantum integrals of motion of the theory to determine, for the two operator families of the model and for each level up to 7, a number of independent form factor