ConwayLife.com

There are many oscillators in this rule, but I want to know if I can find gliders or moving patterns that do not explode. Is there a way to know if they do exist?

Can you describe what sort of behaviour this rule is meant to have? I can't see what the idea behind this rule is, but that may be because there seem to be a few problems with your rule table.

First off, there's a typo in neighborhood:Moore

Second, (for convenience) if you prepend to the table it's easy to automatically create the rule file in golly by simply pasting the contents in like any pattern.

Now for the actual rules:
There are quite a few redundant rules. For example:
2,x,x,x,x,x,x,x,x,0 results in 2,c,c,c,c,c,c,c,c,0 being unused and a,n,n,n,n,n,n,n,n,3 only having effect for a=1
b,b,b,b,b,b,0,0,0,0 is the same as b,0,0,0,b,b,b,b,b,0
n,b,b,b,b,b,b,b,b,3 results in 0,b,b,b,b,b,b,b,b,2 being unused

There are quite a few other peculiarities which make me wonder how this rule table was generated.

Rhombic wrote:There are many oscillators in this rule, but I can't find any infinitely growing patterns nor "gliders". Is there a way to know if they do exist?

This is the most confusing part for me. I see almost no activity in the rule and I can't work out what sort of patterns are meant to exist. This is the only p4 I can find: As for your question, based on what I can see in this rule it is infertile, i.e. no spaceships or infinite growth can exist. But some carefully engineered pattern may allow such growth to occur and I haven't tried to analyse the rule table to see if such a pattern could exist. Existence of spaceships also means that there are patterns which die out, and other than isolated dots I can't find any examples. I find David Eppstein's book chapter, "Growth and Decay in Life-Like Cellular Automata" to be a very useful way of approaching this question. It is focussed on life-like rules, but much of it could be extended to multi-state rules like this one. You can read a preprint at http://arxiv.org/abs/0911.2890

Edited, I had copied one that I made earlier... sort of a failed experiment. Now the code is OK.
EDIT:
Here's a good starting pattern that does not explode:

From looking at your revised rule table, I think it's pretty clear that spaceships are impossible in this rule - because contraction is impossible. Consider the live (state 1 and 2) cells in the northern-most row of the pattern. All the rules resulting in death require at least one cell in the next row to the north to be live, but those cells are all off. The same is true for south, west and east. Therefore, the bounding box of the pattern cannot contract in any direction and ships are not possible.