  We study the phase structure of a phantom tethered surface model shedding
light on the internal degrees of freedom (IDOF), which correspond to the
three-dimensional rod like structure of the lipid molecules. The so-called tilt
order is assumed as IDOF on the surface model. The model is defined by
combining the conventional spherical surface model and the XY model, which
describes not only the interaction between lipids but also the interaction
between the lipids and the surface. The interaction strength between IDOF and
the surface varies depending on the interaction strength between the variables
of IDOF. We know that the model without IDOF undergoes a first-order transition
of surface fluctuations and a first-order collapsing transition. We observe in
this paper that the order of the surface fluctuation transition changes from
first-order to second-order and to higher-order with increasing strength of the
interaction between IDOF variables. On the contrary, the order of collapsing
transition remains first-order and is not influenced by the presence of IDOF.
