Is it possible to have a line rather than a point where the three states of a substance can exist? Most of us are familiar with state diagrams that define which of the three states a substance will take given the pressure and temperature. And that some substances, such as water for example, exhibit the behavior of a triple point - a specific pressure and temperature at which the substance can exist in equilibrium between all three states. The Wikipedia image below illustrates the phase diagram and triple point.

From the phase diagram one can also see that along any of the lines separating the liquid, solid and gas phases (green, red and blue lines) that the substance can exist in equilibrium between any two states.
Imagine though a substance that any one of the red, blue or green lines, is rather a long thin strip in which the third state exists. For example the green line in the diagram that separates liquid and solid would actually be a long barrow strip in which the gaseous state of the substance could exist. Thus you would have a triple point existing along a line at a set of pressures and temperatures.
Are there any substances (pure compounds or mixtures) that might at least approach this behavior?
Can theory predict the composition of substance by which a triple point line might occur. or do fundamental theories forbid it?
 A: No, this is not possible.
First of all, temperature is a particle velocity. (The mass of a molecule can be thought as a constant in this contex)
$$v_{rms}=\sqrt{\frac{3k_BT}{m}}$$
Pressure is basically particle density;
$$p=\rho R_dT$$
So this basically means that if a solid matter is placed on absolute vacuum, then it theoretically holds it's temperature which it had infinitely, while placed inside the vacuum, as there is no particles interacting with it to change it's temperature (particle velocity). In reality it cools down to absolute zero, while releasing randomly single molecules to gas. This same is true for fluid.
But physically seen there is not even true universally defined phase transition between solid and liquid; thus they are often described simply as a "condensed matter". Yet, it's clear that this transition is only happening within this condensed matter due to the increase of temperature in a very certain melting point.   This melting point have basically nothing to do with the process which causes the condensed matter to loose atoms to gas through its surface. The melting point is basically only change in the internal crystal-structure.
This simply means these two possible paths can only meet in one point; Phase transition with in the condensed matter (solid-liquid) and phase transition from condensed matter to gas (kinetic matter).
A: A pure substance can only have a point where three phases coexist.
With two components we can have a whole line of coexistence of three phases. Here is an example you can do in your kitchen: mix water and oil (or heptane or any other hydrocarbon) and bring to boil. You then have three phases: liquid water, liquid hydrocarbon and vapor. The fact that you can do this at various pressures means that the coexistence region is not a point but a line.
