Let Gamma_3 be the closed symmetrized tridisc defined by

Gamma_3 = {(z_1 + z_2 + z_3; z_1z_2 + z_2z_3 +
z_3z_1; z_1z_2z_3) : |z_i| <= 1; i = 1, 2, 3}

Let us denote by b Gamma_3, the distinguished
boundary of Gamma_3 defined as

b Gamma_3 = {(z_1 + z_2 + z_3, z_1z_2 + z_2z_3 +
z_3z_1, z_1z_2z_3) : |z_i|< = 1, i = 1, 2,3}

A 3-tuple of commuting operators (S_1,S_2,P)
defined on a Hilbert space H for which

Gamma_3 is a spectral set is called a
Gamma_3-contraction. In particular, when( S_1, S_2, P) are

commuting normal operators and have their Taylor
joint spectrum sigma(S1; S2; P) contained in b Gamma_3

we call (S1, S2, P) a Gamma_3-unitary. In this
talk, we show that a Gamma_3-contraction does not always have

a Gamma_3-unitary
dilation. In other words, all Gamma_3-contractions do not have normal dilation
to the distinguished

boundary b Gamma_3. To do that, we introduce the
notion of fundamental operator equations for a 3-tuple of commuting

operators (S_1, S_2, P) on a Hilbert space H with
|| P ||<= 1, in the following way:

S_1- S_2^*P = D_PX_1D_P and S_2 - S
_1^*P = D_PX_2D_P , where X_1,X_2 are bounded operators on Ran(I - P ^*P)
closure

and D_P is the square root of (I-P^*P). We show
that the fundamental equations have unique solutions when (S_1, S_2, P) is a

Gamma_3-contraction. For a Gamma_3-contraction
(S_1, S_2, P), we pair up the unique solutions of the fundamental operator

equations as (F_1, F_2) and call it the
fundamental operator pair of (S_1, S_2, P). We
establish the fact that a Gamma_3-contraction

(S_1, S_2, P) has a normal dilation to the
distinguished boundary b Gamma_3 only if F_1^* F1- F_1F_1^* = F_2^*F2 -
F_2F_2^*. At the

end we produce an example of a finite dimensional
Gamma_3-contraction whose fundamental operator pair fails to satisfy the above
operator identity.