In any triangle whose sides have measures a,b, and c,
the triangular inequality tells us:

a + b > c
a + c > b
b + c > a

Since the triangles here have two sides with the same
measure, we will suppose that a = b and the above becomes

a + a > c
a + c > a
a + c > a

 or

2a > c
 c > 0
 c > 0

So we have 

1)    2a > c > 0

Since the perimeter is 93,

we have

   a + a + c = 93

2)    2a + c = 93

Solve 2) for c, 

c = 93 - 2a

Substitute in 1)

1)    2a > 93 - 2a
      4a > 93
       a > 23.25

and since a is an integer

       a > 23 
        
Solve 2) for 2a

      2a = 93 - c  

Substitute 93 - c for 2a in 1)

 93 - c > c 
     93 > 2c
   46.5 > c

and since c is an integer,

     47 > c

or    c < 47 

 
So we have

     a > 23
     c < 47

Let p be the positive integer amount that a is greater than 23
Let q be the positive integer amount that a is less than 47
 
So 

     a = 23 + p 

and

     c = 47 - q

Substituting those into 2)

    2a + c = 93

 2(23 + p) + (47 - q) = 93

     46 + 2p + 47 - q = 93

          93 + 2p - q = 93

               2p - q = 0

                   2p = q

 So now we have

     a = 23+p 

and

     c = 47-2p

Since c > 0

 47 - 2p > 0
     -2p > -47
       p < 23.5

and since p is an integer

       p < 24

And p > 0 so

   0 < p < 24

So there are 23 values p can take on, any of the integers
which are greater than 0 and less than 24.  That's 1 through
23, so there are 23 possible isoceles triangles with integer
sides that have perimeter 93.

Here they all are:

 p   a = 23+p   b = a   c = 47-2p
----------------------------------
 1.     24       24        45
 2.     25       25        43
 3.     26       26        41
 4.     27       27        39
 5.     28       28        37
 6.     29       29        35
 7.     30       30        33
 8.     31       31        31
 9.     32       32        29
10.     33       33        27
11.     34       34        25
12.     35       35        23
13.     36       36        21
14.     37       37        19
15.     38       38        17
16.     39       39        15
17.     40       40        13
18.     41       41        11
19.     42       42        9
20.     43       43        7
21.     44       44        5
22.     45       45        3
23.     46       46        1

Edwin