  To seek for a singularity free model universe from a perfect fluid
scalar-metric cosmology, we work in the "\emph{Emergent Cosmology}" (EC)
paradigm which is a non-singular alternative for cosmological inflation. By
using two methods including Linear Stability Theory and Effective Potential
Formalism, we perform a classical analysis on the possible static solutions
(that are called usually as Einstein Static Universes (ESU)in literature) in
order to study EC paradigm in a FRW background. Our model contains a kinetic
term of the scalar field minimally coupled to the background geometry without a
potential term. The matter content of the model consists of a perfect fluid
plus a cosmological constant $\Lambda$ as a separate source. In the framework
of a local dynamical system analysis, we show that in the absence or presence
of $\Lambda$, depending on some adopted values for the free parameters of the
underlying cosmological model with flat and non-flat spatial geometries, one
gets some static solutions which are viable under classical linear
perturbations. By extending our study to a global dynamical system analysis, we
show that in the presence of $\Lambda$ with non-flat spatial geometries there
is a future global de Sitter attractor in this model. Following the second
method, we derive a new static solution that represents a stable ESU but this
time without dependence on the free parameters of the cosmological model at
hand. As a whole, our analysis suggests the possibility of graceful realization
of a non-singular EC paradigm (i.e. leaving the initial static phase and
entering the inflation period as the universe is evolving) through either
preserving or violation of the strong energy condition.
